Frigid Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠟⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢻⡟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡄⣿⣤⠀⣿⠀⢀⡐⢷⡾⠃⣀⠀⣤⠿⣦⣤⠾⠷⣤⣴⠿⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃⣿⡀⠀⣿⠀⢨⡷⢾⡷⢾⣏⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠛⠀⣿⠀⠈⠠⡾⠻⠄⠁⠀⠛⣿⠛⠙⢷⡾⠋⠛⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⡶⠀⣿⠀⠀⠀⠀⠀⢶⣤⡦⠀⠉⠀⢠⡾⢷⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠄⣿⣀⠀⣿⠀⠀⣸⣆⠀⣀⣿⣀⠀⣰⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣦⣿⠉⠀⣿⠀⠈⠉⠙⣿⠋⠈⠙⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⠗⠀⣿⠀⢠⣤⣤⠿⣦⣀⣴⠿⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⣿⠀⠀⣿⠀⠀⠸⠃⠀⢈⣿⡀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⣰⡟⠀⢠⡿⠂⢿⣄⠀⠙⣧⠀⠀⠀⠀⠛⠉⠛
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢹⡇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠻⣦⣤⣴⠟⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡖⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡜⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⢠⣾⣿⣿⣯⣽⣿⣿⣇⠀⠀⠙⣛⣻⣿⣭⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢟⡽⠉⢀⣀⠉⢳⣤⡞⠋⣉⠉⠙⢿⣿⣿⣿⡿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⠛⠁⠘⣧⡀⠘⠿⠃⣸⣿⣇⠘⠿⠇⠀⡼⠀⠻⠟⠻⣦⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⡀⠀⠈⣙⠒⠶⠟⣯⡉⠙⠳⠶⣖⠛⠁⠀⠀⣼⣿⡟⠀⢀⠇⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠀⢀⣴⠞⠙⡟⠉⠟⠉⢻⠛⠲⣦⡄⠀⣸⡟⠉⠀⠀⠜⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣰⣏⠀⢣⣀⡷⠚⠳⢤⣎⣀⡰⠃⣸⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⢿⣿⣭⣤⣤⣤⣤⣬⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠉⢻⣿⣿⣿⠋⠀⠀⠀⣬⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣟⣩⣅⣀⠀⢀⣴⣿⣿⣿⣷⡀⠀⠀⢀⣈⣹⣧⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⡆⠀⢸⡁⢸
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠼⡟⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠳⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡏⣿⢻⠻⡟⢻⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⡇⢇⠸⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⠃⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡎⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠸⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡇⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⢀⣾⠛⣷⡀⠀⠀⢀⠀⠰⢾⣷⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠠⣤⣾⠃⠀⠙⣷⣴⠀⠀⠀⠀⢀⠙⣷⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣦⣄⣀⠀⠀⠀⢹⣧⠀⠀⠀⣾⠇⠀⠀⠀⣀⣠⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢄⠨⡙⢛⣿⣾⡿⣾⣿⣄⠀⣠⣿⡶⠿⢿⣿⠛⠛⣅⢠⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⡚⠵⠚⣻⡋⠀⣥⠒⠈⣿⣶⣿⠁⡴⠂⣤⠋⡅⠙⠙⣆⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣶⣥⣤⣉⣀⣀⣰⡿⠈⣿⣤⣹⣠⣀⣬⣤⣶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣷⣮⡹⠿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⡿⠠⡆⣹⣿⣿⣿⣿⣿⣿⣿⣷⣎⠻⠿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠚⢁⣚⠁⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⠀⠙⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠏⣱⣿⢿⣷⡀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣺⣶⣿⣿⣿⣿⣟⢧
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢟⠛⣻⣯⣴⣿⣿⣿⡿⣿⣾⣿⣿⣿⣷⣠⣾⣿⣿⣿⣿⢟⣻⣟⣿⣿⣿⣿⡿⠟⠉
⠀⢸⣿⣟⣻⣭⣤⢼⡾⠋⢹⡇⢿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⡃⣸⣿⡀⠀⠀⢀
⠀⠘⣟⠻⣿⣿⣿⡻⠻⣿⠚⣿⣯⣷⣽⣿⠛⢛⣿⣛⢿⣿⣿⡿⣿⡿⠿⠛⠛⣿⣛⣣⣿⣿⣿⣿⣿⣛⠛⢸⣿⣿⣷⡖⣂⠁⠄
⠀⠀⠛⠛⣿⠛⣛⡛⠋⠹⣷⣿⣿⡿⠿⢻⣶⠏⢁⡽⠟⢻⣿⢰⣿⣇⢸⣶⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⢿⣿⣯⣾⣿⡶
⠀⠀⠀⠐⡁⣦⣽⣥⣤⣤⣿⣿⣟⣷⣴⣿⣷⣶⣿⡄⢿⣿⣿⣎⣿⣿⣿⣿⣿⣿⡿⢿⣿⠯⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠘⣻⣿⡟⣿⣿⢙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⣒⡐⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⣠⢤⣾⣿⣿⠗⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠐⣿⡇⠐⠿⠛⣩⠿⠛⢡⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣇⣱⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠑
⠀⠀⠀⠀⠀⠀⠀⠃⠐⠊⠁⠀⠀⠈⠉⠉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣋⣉⣩⠿⠿⣿⣿⣿⣟⡛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠙⠉⠉⠉⠉⠁⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠓⠒⠐⠃⠘⠛⠀⠋⠛⠛⠛⠘⠋⠛⠃

⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣼⣿⠿⠟⠛⠋⠉⠉⠉⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⣤⡀⠀⠀⠀⠸⠿⠟⠀⠀⣀⣀⠀⠀⠀⠿⡿⠃⠀⣀⣀⣰⠟⣡⠟⠁
⠀⢀⣀⣀⣀⣀⣀⣰⣿⣿⣴⠾⢛⣿⠟⠀⣠⣾⡿⠃⢀⣴⠟⢛⣯⡄⢰⣿⡟⢀⣴⠟⠋⣿⣿⠞⠁
⠀⠚⠛⠛⠛⢛⣿⣿⠋⠉⠀⢰⣿⣿⡶⠟⣿⣿⣷⡶⢿⣷⣴⡿⠙⢑⣛⣻⡶⢿⣿⣤⣾⣿⣧⣤⡶⠂
⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⣠⣿⠏⠀⠀⠀⠈⠀⠀⠀⠉⠉⠀⠈⠉⠁
⠀⠀⢰⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡴⠛⠙⠓⠲⢶⡒⠦⠤⣄⡀⠀⢀⡴⠒⠉⠒⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣉⣁⣾⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠈⠙⠣⣄⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠉⠛⢦⡀⠀⠀⠀⣀⠀⠈⠳⣄⠀⠀⠈⢳⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣀⠀⣳⡤⠤⠤⣍⠷⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⢀⡔⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢙⣟⡁⠀⠀⠀⢸⠆⠀⠀⠀⠀⢳⡀⠀⣯⡠⠊⣩⠔⢶⠟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠉⠉⠉⠙⠛⠓⠒⠒⠦⠤⠬⣷⣤⣼⠉⠃⠀⠀⢀⠞⢦⡀⠀⠀⠀⠀⢳⣠⢇⡴⠋⠀⠀⠘⣯⠥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢿⡦⣤⠤⠖⠋⠀⠀⠙⢦⡀⠀⠀⠈⣿⠋⢀⣀⣀⣀⣀⣹⣍⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠋⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣰⣚⢕⣛⣩⣭⣀⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠔⠠⢂⣢⡴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⡉⢭⣿⣿⣉⡀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠻⠥⣄⣈⠙⠦⣄⠀⠀⠀⢀⣠⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠾⠥⠤⠤⠤⠾⠃⠉⢲⡌⢷⣀⣴⠻⡝⢦⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⣀⣤⡴⠒⠋⠉⠐⠒⠒⠲⡦⠤⠀⠀⠈⡟⠉⠀⠀⢹⡎⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠖⠋⢹⡋⠀⠘⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⣠⠎⠀⣿⠀⠀⠘⣇⠀⠀⡰⠀⠀⠀⢸⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣀⣀⣠⡞⠁⠀⠀⠘⣧⠀⠀⢺⢆⣰⠃⠀⠀⠀⢸⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢀⡴⠊⠉⠉⠂⠀⠀⠀⢹⡆⠀⠀⢣⡃⡀⠀⠀⠀⡧⣄⣈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠉⠒⢻⡈⠺⠂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⠀⠀⠀⢠⠁⠀⠟⠀⠀⠀⢣⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠒⠒⠒⠒⠻⢬⣁⠀⠀⠀⠀⠘⠒⠲⠤⠤⢤⣤⣄⣀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⡀⠀⠀⠀⠀⠠⠤⢤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⡞⠁
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣠⠶⠒⠚⠛⠓⠒⠒⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢠⢾⣿⡢⣄
⢰⡏⠉⢉⣲⡶⣧⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠱⡝⢮⡳⡄⠀⠀⠀⣠⢤⢤⡀⠰⣛⡆
⠻⣝⠾⣍⡉⣇⣇⣹⠓⠦⣄⣀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠹⣽⣆⠀⠀⢧⣀⡼⢁⡀⣄
⠀⠈⠙⠲⢭⣙⡚⠧⠤⣤⣀⣉⣉⣳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⣸⢞⡷⢘⣿⡄⠀⠀⠀⢰⣋⣀⣸
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠤⠤⠤⠴⢷⡏⠙⠋⣯⡇⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠞⠓⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣠⡾⢷⡄⢠⡾⠷⣄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠷⠟⠁⠈⠻⠾⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠹⣦⠀⠰⠟⠉⠉⠈⠉⠻⠆⠀⣴⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠋⠉⠀⠘⠳⣤⣀⡀⠀⠀⠀⠀⠰⠞⠃⠀⠉⠙⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢁⡶⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠙⢷⣴⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠙⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠈⠀⠀⢀⠀⠀⠉⠈⠓⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡉⠠⢀⢀⣀⠀⠀⠉⠀⠀⠀⠀⣤⢀⣠⡌⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⠎⠀⠀⠀⠛⠁⠈⠉⢠⡤⠃⠀⠀⠸⠈⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣀⠠⠎⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⢀⣀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠧⠄⠐⠈⠉⠐⠒⠦⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⣰⠟⠉⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠤⠐⠙⡆
⠀⠀⠀⠀⢰⠃⢠⣴⣶⡄⠀⣲⠀⠀⣀⡞⡒⠉⠷⠤⠐⠞⠀⠀⠀⠀⠀⠁⠈⠁⠀⠉⠁⡆⠀⠀⠀⠀⢣
⠀⠀⠀⠀⡎⠀⣾⣿⣿⡏⠀⠀⠀⠉⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣀⣀⠼⠶
⠀⠀⠀⠀⠓⠠⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡠⠤⠤⠤⢤⡤⠦⠦⠰⠤⠴⠖⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠤⣀⣀⣀⡤⠖⠚⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡆⢀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡾⠃⠁⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠠⣄⢀⠀⠰⠇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠈⢷⣦⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢀⣤⣾⡿⠛⠉⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠐⠛⠒⠒⠶⠾⠿⠀⠠⣶⡿⠏⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡀⢈⡀⠉⠀⠸⠁⠀⠀⠐⠒⠒⠂⠀⠀⠤⠄⢤⣀⠀⡀⠀⡀⠈⠉⠁⠀⠀⠀⠐⠒⠂⠀⠀⠐⠗⠠⠀⠀⠀⠄
⠀⠀⢀⠀⠀⣰⠆⠀⠀⠀⠀⠰⠀⠀⢰⡉⡇⠀⣤⠀⠀⠀⠀⠀⠠⠀⠀⢠⠆⠀⠀⣴⠁⡄⠁⢠⠿⠉⣹⣿⠓⢒⠒⠒⢲⣤⠄⠀⠀⠤⠄⠀⢀⣀⡀
⠀⠐⢆⠠⠞⠃⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⣀⢸⡏⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⢣⠞⢀⠴⠰⠶⠛⠁⠘⣤⠋⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣨⡼⠃
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣧⢽⢸⠇⠀⡀⠀⣶⠀⠠⠀⠀⠐⠀⠀⣀⡗⣤⣀⣄⠀⣷⣷⡌⢀⡿⠉⠁⠂⠀⠀⡇⠀⠀⠀⢠⣾⣿
⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⡌⢸⣇⡘⣿⠀⣿⠀⠀⠀⠀⠀⠀⢴⣿⣯⢟⣿⡋⠀⠀⣍⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⡀⠁⢸⣇⢸⡇⠁⣿⣤⣿⠀⠀⠀⠀⠀⢈⣾⣿⣿⣶⡿⢿⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽
⡉⢠⠄⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠅⢠⠀⣿⣼⡇⠀⡿⠉⣿⠀⣦⠄⠀⢘⣻⣿⣿⣿⣿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢰
⠹⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣿⣿⠀⠀⡇⠀⢹⡀⢹⠀⡆⢸⣻⣿⣿⣿⡿⠋⠀⠀⠀⣴⣛⣦⠤⠀⠀⠀⠑⢦⠀⠀⠀⡇
⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠃⠀⠸⡇⠈⠀⡗⢸⢹⣿⣿⡿⠁⠀⠀⠀⣸⡿⠿⡹⠂⣀⠀⠀⠀⠀⢷⡀⠀⠿⠀⠀⠀⠀⡄
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠈⣿⠀⠀⠀⠀⠀⣷⠀⡄⠀⣾⢸⣿⡿⠁⠀⠀⠀⠀⠈⠀⣤⠀⢀⡌⢀⡄⠀⠀⠀⠰⣄⣘⣗
⠴⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠀⡿⢼⣿⠃⢀⡴⠋⠀⠀⠀⠀⣤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⣰⠀⠀⠀⠀⢘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⣴⡶⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣾⠆⣰⣿⠓⡀⠃⠀⠀⠈⠀⢀⡹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡏⠀⠉
⣄⠀⢡⣾⣿⣟⢠⣿⡖⢠⣤⣴⣾⣿⣿⡧⠸⣧⣠⡄⠀⠀⠀⠀⠀⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠰⠛⠁⠀⠀⢸⣿⠇⢸⣿⣿⡿⣿⡿⢃⠀⣿⠙⣿⣆⠀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⢀⣼⣿⣶⣤⠂⢸⣿⢡⡸⣿⣿⣿⠀⠙⠀⠀⢿⣿⣄⣻⣶⣇⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡇
⣾⣿⣿⣿⡏⠀⠈⣿⣾⣿⣿⣿⣿⠀⠀⠀⡄⠈⠛⢿⣿⢿⣿⣧⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢘⡇
⣿⣿⣿⣿⠁⢠⣿⣽⣿⣿⣿⡿⠋⡀⠀⠀⠡⠀⠀⠀⠁⠈⠙⢿⣿⣷⡌⠀⠀⠀⠀⠀⠒⠂⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣸

⣿⣿⣿⡏⠀⢸⣿⣷⣝⠿⠋⡤⠀⠈⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⢶⡄⠘⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡟
⣿⣿⣿⠁⠀⡿⠁⠙⠋⢀⣤⣿⣦⡈⠀⠀⠀⡄⠁⢠⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠰⡀⠀⢈⠗⢴⡀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⣾⣶⠟
⣿⣿⡏⠀⣰⠗⠀⠀⠀⠈⠻⢿⣿⣿⣆⠀⠀⡀⠀⡎⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠛⢺⡃⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣠⣤⣤
⣿⡟⠀⢀⣻⣤⡀⠀⠀⠀⠀⠰⢿⣿⡿⠀⠀⢿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠹⣾⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠉⢻⣷
⡟⠀⠀⣼⣿⣿⣿⣆⢈⣀⠀⢀⣼⠟⣧⣄⠀⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣾⠫⣠⡄⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⢠⡿⠖⠀⢸⡏
⠀⠀⢰⣿⣿⣿⣿⣿⣾⡏⢰⠿⢥⠊⠉⠻⣧⣼⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣠⡿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠋⠰⣄⢉⡀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠛⠉
⠀⠀⠀⠹⣿⣿⣿⣿⡿⢁⠤⠊⢀⣨⣭⣴⠟⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣀
⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⡀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠄⣿⠀⢰⣄⡦
⠀⠀⠀⠀⠀⠸⣿⡷⠀⢊⣥⣿⡿⠋⠀⠀⠀⢀⡀⢂⠰⣇⠀⠀⠀⠀⠀⠀⢀⠇⢈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠁⢁⣶⣿⢀⣾⡇⠁
⠀⠀⠀⠀⠀⢰⡟⣠⣴⣾⣿⣿⣧⡀⠊⠁⠈⠙⠀⣘⠀⢷⠀⠀⠀⠀⠀⢀⡾⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣺⠟⠁⠀⢀⡾⢁⣿⡞⢹⡇
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⡀⢸⠀⠀⡄⠀⠀⣸⠇⢠⣿⠄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⠀⠀⣾⢧⣼⣿⠁⢸⣇
⣀⡀⢀⣾⣿⣿⣿⡿⠛⣿⣿⡟⠀⠀⠀⠤⢤⣤⠀⠀⠃⠀⠈⡸⠁⡆⢰⠃⢠⣿⣿⡆⠀⠀⠂⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⢰⣿⣿⣿⡏⠤⠼⠟⠃
⡐⢀⣨⣿⣿⣿⠋⠀⠄⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⡆⠀⠀⣿⠇⢀⣾⣿⣿⠃⠀⠀⠀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠒⠀⠈⠉⠁⠀⠀⡀⠀⠐⠆
⠘⣌⢿⣿⣿⠇⠀⠀⡀⠀⠘⠻⣦⣄⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠘⢀⣾⣿⣿⡿⠄⠀⠀⠚⠛⠉⠉⠁⡀⠀⣠⠤⠀⢲⣶⣶⣉⠉
⣠⣿⡸⣿⣿⡄⠀⢰⡇⠀⢀⠀⠹⣿⣿⡷⠂⠀⠀⠀⠀⠀⠐⠆⠀⠒⠉⠉⠁⢀⠀⠀⠠⠄⠀⠀⠐⠃⠀⢁⡄⣀⣄⣤⣤⡀
⡟⠉⢧⡸⣿⣧⠀⠻⣷⣴⢻⡇⠀⠙⣯⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠸⠟⠛⠙⠻⣿⣧⣀⠀⠀⠀⠀⠀⠀⡄
⡅⠀⠀⠳⣟⠃⠀⠀⠛⠛⠛⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠁⠀⠀⠀⠀⠂⠀⠀⠉⠻⢷⣀⠀⠀⠀⠀⡀
⠈⠃⠀⠀⠉⠃⠀⠀⢀⡄⠠⡆⠀⠀⠀⢀⣵⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠋⣁⡠⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠰⡄⠀⠀⠈⠻⣦⠀⡀⢠⡇
⠀⠀⠐⠶⣦⡴⠆⠀⢻⡅⠀⠀⠀⠀⢤⣿⣿⣿⣿⣷⠀⠀⢀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠑⠢⣄⢀⡤⠖⠚⠋⠉⠙⠓⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠾⠄⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⡤⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣆⡀⠀⠀⣀⠼⣄⣀⠀⠀⠀⠀⢀⣼⡏⣹⡅⠈⢹⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠚⠉⠁⠀⠀⠉⠉⣭⡟⢦⣬⣽⠝⢯⡿⢯⣿⠛⣾⡭⠶⠤⠞⠵⠯⠴⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⠞⣹⢟⡴⠋⢳⣞⢟⣴⡻⣽⡾⠉⢧⢯⣉⠙⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡾⢧⣀⣨⢗⠼⠒⠷⡼⣅⣀⣼⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⢻⢧⡼⣄⣀⡰⣣⣔⣖⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠫⡟⠧⠼⢻⡤⠗⢘⡎⢳⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠔⣪⠿⠋⠀⠀⠀⠀⠀⠀⣠⠄⠂⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣉⡼⢋
⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠠⠚⠁⠈⡙⠀⠀⠀⢀⡤⠔⠊⠡⠉⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢰⡆⠀⠀⠀⢼⡇⢨⣡⣶⣮⠈⠒
⢠⣾⠴⣻⠇⣸⠀⠀⠀⠀⠚⣍⡄⢦⠀⠀⠀⠄⣦⠐⠀⠀⠐⠚⢻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣾⣧⠀⠀⣠⣿⣿⣈⣋⡼⠟⢶⠶⣶⢀⣆
⣿⡿⣊⢃⣲⣼⣇⣤⡶⠚⢈⠉⢰⢌⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⡆⠀⠀⣫⣜⠅⠀⡜⢛⡎⢉⣻⣀⣤⠈⢃⡀⠙⢿⣀
⣿⠞⣹⣈⡽⢌⣿⣿⡷⠂⠀⠈⡃⠀⠀⢣⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡷⠂⠘⣁⡙⢶⡀⠟⠁⡴⣄⡘⣿⡏⠳⡌⠙⠷⣾⣿
⠿⠓⣅⡟⢘⣦⣹⡿⠗⠚⠶⠖⠉⠛⠧⣌⠳⣦⣦⣴⣀⣴⣠⡠⢼⡿⢄⣀⣰⣄⣦⣦⣾⣷⣶⣏⢜⡁⢸⠞⢉⡅⢸⡡⠒⠉⠻⡀⠻⣌⣗⣦⣤⣀⠈⠻⣿
⣰⡎⠀⣿⢺⣏⣠⡤⢾⣀⣤⠾⢶⢸⡆⠘⠷⣿⣿⣿⣿⣿⣿⡗⡹⡟⢺⣿⣿⣿⡟⢹⣿⣿⣿⡥⢟⠙⠓⠛⣩⢸⢛⣀⠀⠂⣧⠘⣤⣼⣿⣇⠀⠙⠷⢴⣿
⠴⠳⣀⣿⣶⠟⠉⣀⢀⣉⣇⠀⢨⡽⡟⢦⣀⠸⣿⡟⡟⡿⡿⣉⣠⠉⢈⢻⣿⣿⡴⢊⣻⣿⠋⡠⠴⠗⠶⣦⣽⣇⣼⠡⠤⠖⠛⢿⡀⠘⢯⠉⠁⣄⠀⠀⢹
⢀⣶⡟⣹⣷⡞⠋⢀⣬⠋⠀⡖⠈⠀⢀⡀⠙⠳⣾⣿⡏⣿⠯⠎⣀⢧⡼⠨⢿⡵⡁⠊⢉⡟⢋⠀⣠⢤⠦⣀⢹⣿⣅⠀⣠⡀⠀⢹⣉⠉⠺⣆⡀⠈⠂⢤⣀
⣾⣁⣼⣿⡯⠀⠚⢻⣁⣠⡞⠀⢠⡀⠘⢳⡄⣀⠈⡿⢿⡅⠀⠈⡇⢸⠀⠀⠠⡾⠁⠲⢎⠁⣣⣤⡇⢀⡷⣄⣉⣿⣿⠜⠛⣧⠀⢠⡏⠶⣄⡘⢧⣠⠂⠀⢻
⠀⠀⡽⠃⠀⠀⣀⡤⠞⠉⠀⠀⠀⠛⣆⢨⡗⠛⠓⠇⠘⠓⠐⠋⠙⠛⠒⠀⠀⠈⠀⠀⠀⠁⠀⠀⠑⠁⠀⠘⠛⣁⠆⠀⢈⠉⠓⠃⠀⠀⠀⠙⠻⣯⠁⢀⢈
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠲⠀⠀⠀⠙⠻⠇⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⠠⠾⠴⣄⡀⠀⢠⣀⣣⡀⠈⣻⠿⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠈⠀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠐⡀⢀⣠⠀⠀⠀⠰⡷⡄⠀⡀⢦⡀⠆⠠⠆⢈⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⡄⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠁⠀⠀⠘⠃⢤⠍⢧⣿⣧⠉⢻⡄⠀⠶⣿⡿⠈⢣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠒⠲⢛⡃⠒⠠⣄⠀⣺⡿⠂⠘⢧⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠈⠙⣿⣧⠱⣄⠈⣿⣀⠀⢠
⠀⠀⠀⠀⢀⣤⣾⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢈⠁⠀⣀⡀⠈⠂⠂⠖⠈⠉⠑⢻⣿⣄⣿⡀⢹⡶⢠⡯
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⣌⣑⣈⣃⡀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡙⢷⣈⣿⣼⣧⠖
⠀⠀⠀⢹⣿⡿⠟⠁⠐⠿⢷⣮⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⢀⡽⠛⣶⡶⠾⣿⣿⣄⣿⣿⠁⢀⣿⣿⣿⠗⠂
⠀⠀⠀⢸⡟⠀⢀⠀⠀⠀⢠⣿⣷⡀⢸⣽⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⡄⡩⠴⡞⠋⡀⠀⡀⠀⠈⢹⣿⣧⠈⣿⣿⡟
⠀⠀⠀⠿⠶⠒⣿⣧⣾⣿⣿⣿⠿⠇⠜⠻⡄⠀⠿⣿⠟⢿⡿⠿⣿⠿⢛⠛⠻⠋⠊⠟⠀⠂⠒⢿⡐⠛⣺⡟⢿⢀⡏⠑
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣬⢀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠚⠛⠛⠛⠉⠋⠉⠁⠀⠀⠀⢀⣸⣿⣿⣿⡧⠀⢻⠶
⠀⠀⠀⠀⠀⣀⣄⣤⠛⢉⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣄⣀⣶⣿⣿⣿⠿⠟⣿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠉⣉⢉⢛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠀⠀⡉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢶⡀⢀⡼⣿⣬⣀⠀⣤⣈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠘⣷⡟⠸⠋⢨⠉⠿⠈⠉⠙⠿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢻⣤⣾⢋⡀⢡⡆⠁⠀⢀⠈⢀⡔⣰⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠊⠂⣬⠟⡇⢠⢊⠉⠀⡀⠀⢡⣭⡎⢳⣟⠀⠀⢷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣤⠞⢿⣏⠀⢹⣠⣖⡀⡢⢸⡼⢏⣿⡇⠾⣧⢤⣤⣼⠷⣧
⠀⠀⠀⠀⠀⠀⢠⣤⣿⡶⣾⢹⡆⣸⣇⣀⣸⡇⢻⣷⢃⣼⠁⣾⣿⣷⡶⠟⠻⠉⠛⢶⣏⣷⣰⡆
⠀⠀⠀⠀⠀⡰⠿⢸⢏⣤⠿⠿⠛⠿⠾⢷⣿⣷⡿⠙⣦⢿⣶⡿⠋⠁⠀⠠⠖⠃⠀⢀⠹⣮⣟⢳⡄
⠀⠀⢀⣠⣼⣅⣠⡾⠛⠰⠋⠀⠠⠄⠈⠑⠂⢄⠙⠻⣿⣯⣉⣀⠁⠘⠀⠀⠤⣄⡀⠀⠁⠬⠻⢦⣷⣄⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠒⠀⠁⠀⠀⠀⠂⠔⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠻⠼⡁
⠀⠀⠀⠀⠀⠘⡂⢀⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠻⠤⠄⢶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡨⠆
⠀⠀⠀⡀⠀⣠⠂⠀⠶⢷⡝⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣀⠴⠶⢤
⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⣤⣶
⠀⠀⠀⠶⠠⣰⠆⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡁⠀⢄
⠀⠀⠀⠀⠔⠁⠀⠒⣀⠀⠶⣀⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠁⢀
⠀⠐⠧⠄⡤⠀⠀⢀⠙⣀⡀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⢀⠤⠚⠐⠾
⠀⠀⠀⠀⠸⠇⠂⠉⡀⢀⡉⠹⠄⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⠪⣭⣋⠠⣤⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣶⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡎⢇⠀⠀⢒⠄
⠀⠀⠀⡠⠔⠒⠒⢢⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡⡀⠀⠀⢀⠜⠀⠈⢆⠀⠀⠀⣀⠤⣀
⠀⠀⠀⡱⠀⠀⠀⠘⢲⠶⡏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⢊⠀⠐⢳⠂⠁⠐⠲⡛⠀⠰⠁⠀⠈
⠀⠀⠈⢳⠀⠀⠀⠀⠀⡆⢣⣴⠃⠀⡄⠀⡠⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠐⣷⣖⡁⠀⠀⠀⠀⣑⣤⠀⠁⠒⠈
⠀⢠⠤⠶⠀⠈⠁⠈⠄⠉⠁⠈⡆⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⡡⠤⠀⠄⣀⣀⡤⠋⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⢇⠀⢀⡁⠀⠀⠀⠐⠀⠀⢐⡍⡂⠀⠐⠀⣊⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⢈⠽⠋⠙⠀⠂⠠⠐⠀⠛⢍
⠀⢰⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠒⠀⠀⠀⠒⠁⠀⠀⠠⠀⢌⢑⡶⠀⠀⠀⠀⠀⠀⠀⠴⣖⠉
⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠄⠀⠠⠤⠤⢤⠤⠀⠀⠤⠵⠂
⢹⠀⠀⠒⠒⠒⠁⠀⠈⠉⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠤⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠚⠀⠀⠀⡇⠀⡀⢀
⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⣀⣀⠤⠀⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠁
⠀⢧⣀⠄⠀⠀⢣⣀⠄⠀⠀⠀⠳⠤⠂⠘⠦⠄⠀⠘⠦⠔⠀⠈⠳⠤⠤⠂⠈⠳⠤⠤⠔⠊⠀⠳⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢷⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣟⣘⡾⠾⣼⣼⣰⢢⢄⠀⠀⠀⠀⠀⠈⡈⠁⠀⠄⠀⠠⠀⠁⠀⠐⢀
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⢀⠈⠻⡇⢠⣴⠀⠈⠉⠛⠺⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠷⠶⢤⣄⣑⡆⠙⠂⠕⠚⣣⡰⠟⣠⠋⠀⠀⠀⠀⠰⠁⠀⠄⠀⠀⡀⡐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠐⢠⡄⠀⠀⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⢸⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠜⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠘⠆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⡄⠓⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡈
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⡐⠒⠅⠀⠀⡒⠲⡁⠈⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠴⠦⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠒⠶⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢰⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣥⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠸⣿⣿⣿⣿⡿⣱⣮⣝⣛⣛⡛⣫⣭⣶⣿⡿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣩⣭⢩⣵⣾⣿⣿⣿⣿⣿⣿⣶⣭⢉⣭⡁⠀⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣾⡀⠀⠘⣿⣇⠾⠇⠿⠇⠿⢆⣿⡟⠀⠀⠀⠀⢠⡀⢰⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠈⣿⡀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣀⠀⣷⠈⠻⣾⡷⠋⢀⡿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⠈⠛⠇⠀⠀⢠⣭⣭⣭⣥⠀⠀⠀⠀⣉⣻⣿⣇⡀⢸⡇⢀⣼⣷⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⢴⣾⣿⣿⣿⣿⣷⡄⣠⢈⣉⠉⠀⠈⣹⣿⣿⣋⠁⠈⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⠟⣡⣾⣿⡦⢉⣻⣿⡟⠉⢸⡇⠙⢻⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣧⣫⡅⡿⠀⣠⣾⣷⣄⠈⣷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠘⠁⢸⡇⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣾⢃⠀⠀⠀⠀⠀⢀⢸⠀⢠
⠀⠀⠀⠀⠀⠀⣳⣄⠀⠀⠀⠀⣇⢸⡁⠀⠀⣻⣿⡟⣡⣴⣼⣧⠀⠸⠈⣤⢸
⢀⠀⠐⠃⠀⠀⢻⣿⠀⠀⠀⣎⢿⢼⣧⠰⠀⣿⣽⣿⣿⣿⣿⡟⡀⠒⢀⢿⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⡀⠁
⣸⣿⠀⠂⠀⠀⡏⣗⠀⠀⢸⣿⣿⣾⠛⠘⠀⠁⢹⣿⣿⣿⣿⡏⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠘⣳⣶
⣿⣿⣄⠀⠀⠀⠁⠃⠀⠀⠀⢹⡿⠋⠀⠀⠀⢀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠃⠀⠀⣽⡟
⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠟⠋⠁
⠸⠟⠿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⣿⠆⠀⠀⠀⣠⣴⣾⠏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡏⣇⠀⣸⣶⣝⣠⢴⣾⡿⠛⠁⠀⠀⠀⠀⠀⣀⢠⠖⠈⠁⠀⠀⠲⣻⡶⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⡰⢧⠀⢠⡀⠀⣄
⠀⠀⠀⡀⠀⠀⠀⣦⡀⣀⠀⢞⠛⠋⡼⢡⠿⣞⣏⠌⢉⡅⠀⠀⠀⢀⣤⠆⠉⠁⠂⠀⢸⡧⠀⠀⣿⢉⣦⡄⠀⠀⠰⢿⠆⠀⠀⠀⣝⡄⠶⣷⡀⡶⣷⢴⣿
⢠⣦⣸⣷⠀⣶⣜⣍⣁⣿⣶⣯⠼⣷⢶⢿⡀⢈⠛⠁⠈⠀⠠⠔⠪⢿⠇⠀⠀⠀⠀⠀⣻⡟⠄⠘⣻⣾⢧⣀⠀⠀⠶⢻⣼⢠⡄⢸⣏⣹⢼⡀⢻⢧⣼⣌⣿
⢸⣿⣵⢾⣿⣯⠼⡯⣤⢿⣃⡞⠀⣿⠂⡸⣧⣰⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢤⡇⠿⠠⠶⣿⣌⡟⠀⠰⡶⣿⣄⠚⠄⠚⣿⣥⡾⠷⠾⣧⣏⢸⣹
⢸⣿⢿⡤⣿⣁⡠⢆⣦⣤⣿⣧⣾⣿⡿⢀⣾⣷⡀⢦⣠⣀⡆⠀⡀⡀⠀⠀⠀⠀⠐⣸⠟⣿⢼⣤⠚⢻⠉⠉⠀⠠⠖⠻⠎⢃⡄⠘⠻⣯⡴⣦⡀⢹⣽⢻⣿
⢸⣷⣿⣿⡀⣠⠀⣈⣧⣀⠙⣿⣿⣿⣽⣿⡏⣿⣷⣾⣿⣿⣿⣿⣿⡿⣿⠶⣾⣶⣋⢼⡴⢿⠶⣴⣦⣼⣤⣤⢺⣧⣤⡖⣲⠃⠑⣊⠻⣿⣾⣿⣿⣿⣷⣼⣿
⢸⣿⣿⡿⠉⢉⡶⠏⠈⢻⡟⢻⣿⢿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⢷⣷⣯⣿⣿⣿⡵⢚⣇⢘⠧⣜⣦⣿⡿⡄⣩⡐⢿⣋⠉⡇⢸⡅⢙⣻⣿⣿⣿⣿⣿⣿⣿
⢸⡿⣿⡿⠑⢺⡀⠈⡧⣀⣷⡖⠟⠘⠁⠹⡇⠁⠈⠋⠛⠁⠀⠀⠀⠉⠙⠛⢻⡡⠒⠻⠋⠀⠴⠗⠀⣿⠁⢯⠡⠅⠐⠵⡎⢁⣏⠙⠦⢭⢹⠟⠛⠉⠀⠙⠻
⠀⠀⡈⢛⠚⠉⡉⢁⣄⠀⠈⠿⠒⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠘⠃⠀⠚⠂⠉⠁⠀⠀⠀⠐⠈⠘⠃⠋⠸⠀⠆⠀⠦⠴
⠀⠀⠉⠉⠀⠈⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⢀⡽⢛⣽⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣠⢀⣟⣴⣷⠆⠂
⠀⠀⠈⣵⣟⡇⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⠀⠀⣀⣴⣾⠖⠀⠀⠋⠀⠀⠀⠜⢡⡢⢟⣿⠒⠋⠀⠀⠀⠠⠄
⠀⠀⠀⠐⣷⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣀⣄⣠⣂⡘⢁⣠⠞⠁⢠⡄
⠀⣄⠀⠐⢉⡿⣿⣷⣄⣤⣄⣠⢄⠞⠋⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⢾⠟⠁⠀⢠⣇⠀⠀⠛⣿⢫⣉⠴⢶⠞⠁⠀⠀⠀⠁⠠⣦
⠀⠈⠀⠀⠀⠰⣻⣿⣫⠟⣿⣧⣼⣤⣀⡄⠀⢠⢀⡀⡤⡒⢂⠒⠀⠀⠁⠈⡠⢀⣦⣄⣑⡀⣀⣰⣵⣊⡛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠂
⠀⠀⠀⠀⠈⠀⠀⠌⠴⠚⣉⣔⣿⡿⣠⡐⠛⣌⢿⣿⣇⡇⡘⠦⢠⣶⣶⡈⢫⡾⣟⣭⢿⣿⡿⠛⣹⣶⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠛⠉⢡⠈⠳⠛⢠⡿⡿⠇⠝⠩⢏⣄⠦⠾⣯⣡⣟⡫⠷⠚⠋⢀⣦⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⣠⠔⠊⢀⠐⠁⢤⡃⡴⣼⠋⠀⠀⠀⠚⠉⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣼⠂⡝⡡⠋⠻⠽⠹⡒⢄⠈⢤⣿⠀⠀⠀⠂⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠋⢀⡄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢠⡄⠸⠇⠠⠃⠃⢨⠀⢸⡍⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⢀⠀⡑⢮⠀⡀⠑⡞⠁⣻⠔⠁
⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⣄⢧⠰⠶⠖⠶⢀⡎⠀⠼⡅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣈⡧⠷⣃⡄⡗⠁⣀⠌⣄⠈⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠚⠣⡜⣶⣂⠀⠀⠜⢢⠀⡶⠉⠀⠘⠛⠀⠀⠀⠀⠀⠀⡠⠾⠋⠉⠂⠙⠀⡇⢰⠋⡠⠅⣶⠋
⠈⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠃⡄⠁⠈⣻⡀⠓⣶⣼⣹⠀⠄⠀⠠⠠⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⠀⠀⠀⣀⣀⢤⡤⢅⡦⣴⠀⠀⠰⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣶⣿⣷⠀⢈⠋⢻⠛⠲⠃⠀⣑⢠⡶⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⣭⢀⠃⠀⠀⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠑⢪⠀⣀⡀⢀⠦⠊⠉⠉⠋⠉⠀⠀⠀⠀⢤⣄⣀⣀⣀⠀⠀⠀⠀⣤⣀⣘⣿⢀⠀⠀⠤⠀⠈⠁
⠀⠀⢀⣀⠀⠀⠀⠘⠋⠉⠉⠀⢰⣂⡀⠉⠀⠀⠄⠽⠥⠬⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⣺⣿⡿⠇⠀⠀⠠⠄⠀⠙⠛⠓⣺⣿⣿⠹⠤⠛⠒
⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠄⠈⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠛⠀⠈⠉⠁⠐⠉
⠀⠀⠀⠀⠀⠐⣀⠐⠂⠀⠀⠀⢀⡢⠀⠀⠀⠀⠀⠀⠑⠂⠀⠁⠀⠈⠉⠉⠁⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣄⣀⡀⠀⠀⠀⠢
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⢯⠧⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣵⣶⣤⣾⣵⣿⣽⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⢋⡀⠅⣨⣿⣿⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢠⣿⣤⣼⣮⠁⠀⠀⠈⣻⠃
⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣴⠃
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀
⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠙⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿
⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⠀⢀⣴⣶⣦⠀⡇⠀⡠⠊⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠒⢏⠀⠀⠀⠀⢅⠀⠸⡟⠋⠉⠉⠁⠒⠿⣿⣷⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛
⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠦⠤⠊⠁⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⡈
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠉⠒⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⠀⡇
⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣀⡀⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⡤⠀⣀⣀⠀⡀⣤⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢻⡀⣀⠄⢻⡟⡼⠀⠀⢠⢸⠁⠘⠁⢀⠀⠀⠐⠌⠀⣷⠀⡏⢆⠰⡅⣿⠘⠇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⡎⣧⠇⠀⢸⣧⠀⡐⠋⠀⡘⠀⠀⢠⠎⠀⠀⠀⠘⣀⣷⡟⡁⠁⠀⢠⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠀⢹⠀⠀⢸⣿⠀⠁⠀⠀⣧⡤⠰⠋⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⠀⢹⠆⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣆⠀⢸⣿⠠⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠘⣋⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢽⠀⠀⡀⠀⠀⠀⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡷⣴⠒⠛⠁⠀⠀⠀⠀⢀⣠⣶⣶⣄⣸⣠⡾⣶⣦⡀⢰⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⠇⠀⠀⠀⠀⢀⣠⡶⢿⣿⣿⣿⣿⣿⡛⣅⠉⠙⠃⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡇⠀⠂⢀⣠⡶⠟⠉⠀⠀⠙⠻⣿⣿⣿⡇⢸⣿⣦⡀⠘⡇⢏⣀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠉⠷⣿⣿⣇⣤⠞⠋⠅⠈⠀⢀⠀⠀⢠⠠⡄⡙⢿⡟⢸⣿⣿⣿⣷⣧⡇⠸⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⢀⣠⢼⣿⠛⡇⠀⢀⡶⠒⠲⠓⠐⠁⠀⠀⡗⣱⢾⣏⡼⣿⣿⣿⣿⣿⣿⠦⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⣠⠖⠋⠡⢼⡿⠀⢃⡴⠻⣤⠀⠀⢠⠀⠀⠀⠀⠀⢹⣾⡇⠃⠀⢙⣿⡟⣿⢶⣟⣿⡄⠆
⠀⠀⠀⠀⠀⠀⣤⡴⢞⠋⠀⠠⡄⡀⢸⣷⠓⡟⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡿⢀⣴⣿⣿⣿⡟⢸⣿⡟⡃⡄
⠀⠀⠀⠀⠀⠀⠈⠀⢻⠀⠂⠐⠛⠙⢾⣿⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢾⣿⣿⣿⡏⡧⢸⡽⡆⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⢸⣿⠇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡟⢹⣿⣿⣿⡇⣷⢸⠇⣣⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣠⣾⣿⣷⡺⢿⣿⣿⣧⡏⠘⡇⣏⢸
⠀⠀⣤⣄⠴⣶⣼⣿⣾⠀⠀⠀⠂⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⣾⣿⣿⣿⡟⡁⣾⣿⣿⢿⡇⣠⠃⣿⢸⣦
⠀⢸⣿⣿⡿⣿⣟⣿⣿⠀⠀⠀⠀⠁⢸⣿⣿⣅⠀⠀⠀⠀⠀⡄⠀⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢸⡗⣿⠀⡏⢸⡷
⠀⢸⣿⠿⠷⡯⠿⡿⠿⠀⠀⠀⠀⠀⢸⣿⣇⠋⠀⠀⠀⠀⠀⠁⠚⠻⡿⡻⠛⠟⢿⣧⠊⠟⠿⠿⢸⣷⡿⠟⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠀⠀⠘⠛⠁⠀⠀⠀⢸⠋⠁⠀⠀⢰⡇
⠀⢀⠀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣄⡤⠼⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠚⡇
⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠩⠉⠀⠉⠉⠉⠉⠉⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⡇⠀⠐⠒⠀⠀⠠⠀⡁⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⠀⢀⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⡇⠀⢀⢀⠀⣀⣀⠀⡅⠀⠀⠀⠠⠎⢁⠀⠠⠀⠀⠀⠐⠐⠨⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠇⢀⡀⢐⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡇⠀⠀⠀⠀⠘⠂⠀⡆⠀⠀⠀⠀⢃⠈⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠤⠄⡇⠀⠠⠤⠀⣠⡄⠀⡁⠀⠀⠀⠐⠀⠓⢄⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⢀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠛⠒⠋⣀⠈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠐⠀⠀⠇⠀⠐⠀⠀⠤⠆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠊⠁⠅⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⡀⠀⢀⣀⠀⢀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢻⠀⡆⠀⠣⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠉⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢡⡁⠠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠀⠀⠀⠅⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠠⠤⠄⠇⠀⠠⠤⠀⣠⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠈⠉⠁⡁⠀⠈⠉⠀⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣃⡀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣽⡇⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⡇⠀⠐⠀⠀⠒⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣷⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⠿⠿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢃⡈⢉⠈⠀⠀⢄⠀⢱⣆⣐⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢢⣌⠀⠀⠀⢋⣁⠼⠒⡉⠤⠖⠨⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢁⠀⠀⢰⡈⠠⡴⠂⠉⢀⠀⠀⠈⠙⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⢠⣔⣢⡄⢈⠁⢩⠀⠀⠀⢆⠀⠅
⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠀⠐⠁⠈⡀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠐⠒⠒⠂⠒⠀⠂
⠀⠀⠀⠀⠐⠠⠀⢀⠀⠀⣀⡔⠀⢈⠂⠀⠀⡠⠈⠁⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠈⣀⠉⠀⣈⠈⠹⠀⡴⠁⡂⠀⠀⠀⢀⡀⠘⡄
⠀⠀⠀⠀⠀⢣⡈⠀⠀⠉⡀⡀⠉⠀⠉⠀⠆⢊⠐⠋⡂⠀⠈⠒⠀⠅⠀⠀⢀⠄⠀⠤⡀
⠀⠀⠀⠀⠀⡀⢱⢔⣄⣀⣀⡀⠀⠀⠀⠀⢃⠀⠀⠈⠃⠀⠰⠆⠀⠀⠀⣰⠃⠀⠀⠀⢈⡄⡀
⠀⠀⠀⠾⠀⠀⠤⠱⠤⡄⢼⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠍⠀⠀⡰⢃⠊⢐⡀⠠⠁⠂⡍⠂⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⢋⣖⠀⠀⠀⠀⠀⠰⠨⠬⠉⣂⢝⠡⡄⠸⣄⡐⠭⠔⢀⣀⠂⠀⢀⠀⠀⠄
⠀⠀⠀⣠⢀⠅⠄⠉⠈⠗⡧⡅⠀⠀⠀⠀⢀⣣⠁⠀⠤⢄⢀⢑⣷⡴⠤⠈⠀⢠⠤⠤⢆⡠⠶⠤⠘
⠀⠀⠰⠧⠀⠀⠑⠂⠈⠠⠁⣈⣄⠀⠀⠀⠀⠉⡄⠀⠀⠊⡀⠀⠈⠀⠀⠐⣠⢦⡤⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠂⠀⠀⠀⠀⠀⢨⡉⣩⣉⡅⠀⠠⡀⠀⠁⠀⠉⣕⣾⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠿⠄⠇⠀⠀⢀⢠⠀⠀⠀⠀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠁⠀⠀⠰⠄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠆⠀⠀⡤
⠀⢀⡄⠀⢀⣄⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⡠⠀⠠⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⠉⣄⠑⠀⠁⢀⠀⠀⠀⠀⢀⣿⠀⠐⠀⠀⠀⠺⠀⠀⠐⠀⠀⢢⠀⣰⡄⠊⠀⠠⠀⠈
⠀⠀⠀⠈⢀⢃⠈⠀⣠⣤⣤⣵⠀⠀⠖⠢⠂⠒⠆⠀⡌⣂⠀⣐⠩⠁⢀⢠⣆⠰⢃⠱⡀⠤⣼⡇
⠠⣾⡀⠀⠐⠐⠰⠛⠛⠛⠃⠌⠶⠒⠀⠓⠐⠃⠘⠘⡖⠿⠀⠿⠀⠗⠀⣤⠈⠻⠫⠛⢋⠈⢀⠃⣠⣤⡀
⠈⣾⡅⣠⠨⠺⠀⠰⠂⠗⠀⡂⡄⠚⢀⠄⠄⠄⠘⠃⡇⠀⣏⡅⠀⠀⣶⠀⣶⠀⠿⠿⠀⢠⡄⢘⢁⣤⠉
⠐⠛⠂⢸⠐⠾⠀⠐⠂⠂⢸⠽⠃⠀⠀⠂⡀⠂⠀⠰⡇⠰⠃⢧⠈⠘⠛⠀⠛⠂⠿⠽⡄⠘⠃⢸⠀⠅⢠

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢘⣿⣻⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡃⠀⡄⠀⢠⣯⠄⠀⠀⠀⠀⣸⠀⠀⢀⠀⠀⠀⠀⣨⡆⠀⠀⠀⠀⢀⡀
⠀⠀⠠⣿⣿⠿⣿⣄⡀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢢⢠⣇⠀⣰⣿⡇⢰⠀⠀⣴⣿⡷⠀⢀⣆⢰⣿⣇⠀⣾⠀⠀⡀⠀⣼⡆⢀⠀⠀⢀⣼⣧
⠀⠢⢼⣿⣿⣶⣉⡶⢿⣤⠀⠀⣼⣿⣇⠀⢰⠀⠠⣿⠀⠨⠀⢠⣿⣾⣿⣤⣼⣿⣿⣿⠀⢴⣾⣿⡟⠂⢼⣿⣿⣿⣿⣿⣿⣦⣠⡆⠀⣼⣧⣾⡆⠀⢠⣿⣿⠄⢰
⢠⣾⣾⣿⡿⢿⠯⢹⣿⣿⣷⣾⣿⣿⣿⣶⣿⣶⣼⣿⣦⣾⣄⣸⣿⣿⣿⣾⣿⣿⣿⣿⣀⣿⣿⣏⣼⣷⣿⣿⡿⠟⢟⡛⢿⣿⢿⣿⢈⣿⣿⣿⡇⠀⣼⣿⣿⣇⣾
⠘⢿⣿⣿⡿⣌⣴⣿⠿⠉⣿⡿⠿⠿⠟⠛⠙⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⢿⠟⠓⠚⠛⣵⡎⢿⣾⣿⢾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠀⠛⠻⠿⣤⣿⣿⢡⠤⡤⠠⠀⡦⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠉⠀⠂⠀⠀⠀⠀⠀⠉⠋⠉⠉⠙⠛⠛⠿
⠀⢺⠙⠇⠉⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⣤⣤⣀⣀⣠⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⡄⠤⠀⠀⠦⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣶⣶⣤⣄⣀⣀⣀⣀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⠉⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⣀⠤⠈⠀⠀⠑⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢿⣀⣀⣀⠀⠄⠈⠁⣠⠤⠀⢠⡞
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠁⢐⣄⢘⣿⣿⣇⣀⠄⢀⡀⠀⢀⣰⠎⠀⢄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠃⠀⢐⠉⠉⠉⠛⠻⢿⣿⣀⣈⣀⣠⣿⡿⠎⠀⢸⠀⠀⠀⠀⠀⠀⣀⡀⢀
⢀⡀⠒⠀⠀⠀⠀⠀⠒⠀⠌⠀⠀⠀⠠⢕⡄⠀⠀⠐⠀⠈⠉⠙⠛⠉⠅⠀⠁⠠⠊⠄⡀⠀⠀⠤⠟⠛⠁⠀⡄
⠀⠳⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣀⠘⠐⢿⡦⠀⠈⠢⡀⠀⠀⠀⠀⠃⢰
⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⢠⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢧⡔⠀⠀⠈⡄⠀⠓⣻⡀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠊⡴⠁⠀⠀⡯⠀⠀⠀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠢⠖⠉⠙⢦
⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣤⠤⠤⠶⠾⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢉⡒⠃⠒⢤⠀⠀⠈⠑⣦
⠀⠀⠀⢀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠇⠀⠀⠀⠸⡃⢠⡀
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⢀⣀⡴⠂⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡍⠁⢀⠀⠀⠙⢦⣀
⠀⠀⠀⢡⠔⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠵⢦⡀⠀⠈⠳⣤⡀
⠀⠀⠀⢦⠀⠀⢀⡤⢊⠀⠀⠀⠹⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠙⢦⣄⠀⠀⠙
⠀⠀⠀⠀⠙⠛⠁⠀⠸⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠓⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠙⣆⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠘⣆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢦⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⡤⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠈⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣀⣀⣀⣠⣤⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢰⠏⠂
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠂⠀⠀⠠⠒⠀⠚⠀⠀⢀⡠⠀⠂⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⢸⡆⣰⣧⣆⠒
⣴⣧⣿⣇⡆⠀⠀⠘⣯⠐⠂⠀⠀⢲⡄⠀⠀⠂⠻⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢰⢷⠀⢠⣾⢷⣭⠌⠛⢶⠒⢠⡆
⣻⡟⢫⣿⣧⣷⣏⡠⠀⠔⠐⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡔⢯⡀⢘⡁⢂⠪⡦⠴⡘⢦⣝⢳
⠚⣯⡶⢹⣿⣿⠋⠂⠀⠂⠐⡈⢺⣶⣆⣤⣀⣤⢀⠀⠀⠀⢀⡀⣤⢀⣠⡊⠃⣴⠞⠃⢜⡺⠤⠉⠱⣿⠀⠈⠀⠙⢿
⢈⠁⠖⢸⣉⡤⠎⣠⠆⠀⠀⠐⠲⣿⣿⡿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠈⠈⠺⣂⠘⣯⡤⠒⣀⡠⣜⡀⠀⠈⠶⢾
⠸⠄⢠⣟⡉⠠⠠⠶⢀⣰⢿⠑⢠⣙⣯⢓⣹⣭⠟⣋⡁⠀⠀⠀⠙⢿⠀⠈⠙⠳⣎⢢⡟⠒⠉⠈⠣⠌⡂⠀⠠⡀
⠀⠀⢠⡏⠠⢰⠂⢠⠊⠙⢄⣯⡀⠀⠉⠁⠀⢄⣪⣷⣻⣓⣦⣴⡶⠂⠀⡀⠁⣌⠚⢿⣗⠄⣇⠀⢰⡆⠈⠲⡀⡀⠑
⠀⡰⠋⠀⢀⡬⣟⠀⠀⠀⢣⣀⠧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⡋⠀⠀⠉⠐⠉⠄⠀⠁⢿⡁⣀
⠀⠀⠀⠈⠀⠀⠀⠙⠊⠀⠊⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠘⠉⠀⠄⠐⠤⠦⠀⠽⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⠿⠛⠛⠛⠛⠿⢿⣛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠉⢀⣶⣌⡙⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠈⠛⠛⠿⠷⠄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠉⠐⠃⠘⠀⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣬⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⠿⠂⠀⠀⠀⠀⠀⠀⠐⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣤⡴⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠳⠶⣤⣀⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⣀⣠⣤⡴⠶⠚⠛⠻⢧⣤⣀⡀
⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠷⢤⣄⣀⠀⠀⣠⣶⣶⡄⠀⠀⢠⠶⢿⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡶⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠐⠂⠐⠀⠂⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠰⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⡀⠀⣄⡠⠀⢤⢖⠀⢄⠀⠀⢰⣷⣀⣰⣸⡍⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠉⠚⢩⡅⠀⣤⠀⠑⠈⢛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⡴⠋⠉⠒⠛⠛⠃⠀⠀⢨⡿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣄⠈⢂⠀⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢁⠀⠒⠂⠐⠂⠒⠒⠢⡈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢉⡷⠲⢀⠿⡿⢀⣶⣐⡖⠲⢄⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢈⢠⡾⠆⣀⣀⠀⡇⠀⠀⣣⣠⡝⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⣘⡇⠀⠓⠈⠉⠉⠙⡀⠀⠀⠀⠂⢀⠀⣀⠀⠉⠢⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⡀⢀⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠉⠰⡮⠀⠀⠀⠀⠀⠉⠑⠀⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠁⠋⠁⠀⠀⡌⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⠀⠀⣠⠎⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠶⠦⠤⠐⠒⠚⠉⠉⠀⠻⣄⡀⠀⠀⠠⡄⠀⣠⣤⠞⠋⠀⠄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⢳⠠⠉⠀⠀⠀⠁⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠴⠒⢦⠀⣧⠰⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠸⡆⢿⣆⠀⠘⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣀⣀⣹⡀⢻⣆⣀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠉⠉⠀⠀⠬⠣⢀⡿⠙⢧⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠉⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠒⠈⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠓⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠀⠐⠘⠂⠀⠀⠀⠀⠆⠀⠘⠓⠀⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢎
⠁⠀⠀⠀⠀⠈⠓⠋⠉⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠋⠁⠈⠀⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢤⡤⢀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣤⡤⠄⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠙⣷⣿⣂⣀⣀⠹⡆⠋⠀⠁⠀⠉⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢹⡄⠀⠀⠀⠀⠀⠂⠈⠁⠘⠉⠓⠐⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠁⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠀⠀⠀⠀⡀

⡀
⠃

⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢀⣨⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣫⠄
⠀⠀⠀⠀⢠⣄⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣩⢧⠶⠂
⠀⠀⠀⠀⣄⣈⣱⣆⣠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣤⣩⣿⣦⣶⠛
⠀⠀⠀⢠⡌⠉⢡⣎⣁⣀⡄⠀⣠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣉⣤⣾⣥⡼⠳⠶⠾⠃⢀⠀⣠⠄
⠀⠀⠀⠀⠓⠚⣯⠉⢉⢁⣠⠀⠋⡀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠀⡞⠀⠀⠀⢰⠃⠁⢺⣤⠿⢶⣧⡾⢿⣶⠶⠾⠋⠊⠀⠟
⠀⠦⠐⠶⡴⠾⢛⡓⠟⠉⠁⠀⠀⣧⠀⡀⢀⡀⠀⠈⢧⡄⠠⢠⠆⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣟⣫⡿⠶⢟⡿⠶⠶⠂
⠀⠀⠶⣦⣀⣴⠞⠛⣶⣄⣼⠻⠇⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣳⣫⣿⣭⣿⣟⣻⡷⢶⡾⠇
⠀⢀⣤⣬⣡⣂⣴⣾⣇⠈⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣋⣽⠶⠖⠺⢛⣿⣷⣤⣤⣶⠄
⠀⠉⠉⠉⠙⡉⠉⣹⡋⠛⠋⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠶⠼⠿⠿⣿⣧⣤⣤⡤⠆
⢀⡝⠛⢿⣍⣠⠾⠻⡿⢷⠆⣦⠆
⠈⣿⠏⣉⣈⠈⣀⢾⣧⣀⠀⡾⠟
⠀⠀⠀⠀⠿⠋⠏⠈⠙⠛⠋⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⡟⠄
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⢴⠋⠀⠀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⠀⢴⢎⡠⢤⡀
⢠⣆⣼⡗⣠⠀⠀⠀⠴⠂⠠⠀⠀⠈⢀⡀⠀⠀⠠⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⢀⣼⡆⠛⠛⠿⣤⣠⡄⣀
⣿⣿⡟⣷⣿⣠⣠⠔⢐⠁⡄⣀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⣆⠀⠀⣯⠃⠀⡜⢛⠉⣻⡁⣀⠹⣀⠠⣿⣀
⣸⡿⢀⢻⣿⣿⣿⢞⠁⠈⡁⠀⠡⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢀⢘⣤⠓⣄⡔⠀⠶⢌⣯⠈⠣⠈⢳⣮⣿
⢨⡿⢛⢰⣿⠟⠃⠌⠁⡀⠀⠀⢂⢘⣿⣷⣿⣶⣼⣤⣄⣀⡀⣠⣤⣤⣷⣤⣼⠴⠅⣘⣋⠀⣀⡥⠋⣁⠀⠈⢿⠀⠀⠠⡀⠈⣿
⢠⡀⠊⢈⣥⠖⠋⠀⠚⠀⠀⣄⡀⠑⢿⣿⣟⣹⣽⣽⣿⡿⠟⠟⠿⠿⣿⣿⠟⠀⡀⣈⡑⡆⢈⡶⢋⡠⢦⠜⢦⡂⠀⠀⠉⠙⢿
⠘⠋⠀⢯⡥⠀⠁⠚⠁⢤⡜⢳⡀⠱⠦⢷⣀⢼⠷⠛⣡⠤⣄⠀⠀⠀⠈⠻⠀⠀⣀⢉⠻⣄⣾⢉⢁⠀⠀⡁⠂⢄⡀⠀⠢⢀⠠
⠀⠀⢠⡎⠀⠂⣏⡀⠔⠀⠈⠤⠷⠀⣀⢀⠀⠈⠰⣭⡼⠏⢿⣽⠶⠾⠟⠁⣨⠈⢁⣁⣉⣻⣿⡷⢛⣧⣀⣿⣷⢄⣻⣇⣧⠈⢿
⠀⠠⠋⠀⣀⡤⠶⣅⠀⠀⠈⢳⡠⠲⡍⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⢩⠉⠛⠙⢦⠠⠉⠻⣏⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠠⠀⠊⠁⠐⠄⠀⠠⠳⠄⠸⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣧⡀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⢀⣼⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠙⣄⠀⠀⠀⠀⠀⢟⣆⠀⠀⠹⣾⣷⠏⠀⠀⣰⡿⠀⠀⠀⠀⠀⣠⠋⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣦⠤⡀⠀⠀⠘⡜⣆⠀⠀⢸⡇⠀⠀⣰⢣⠃⠀⠀⢀⠤⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡻⡮⣝⢒⡳⢷⡘⣕⢦⣸⣇⡤⣪⢇⡾⢖⡒⣪⢵⢟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢳⢙⠮⣣⣀⠤⢧⢹⠓⢼⡧⠚⡏⡼⠤⣀⣜⠵⡋⡎⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢷⣒⠦⢼⣎⠀⡜⠛⢄⠀⢣⣧⣸⣇⣼⡼⠀⡠⠛⢣⠀⣱⡧⠤⣒⡾⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⢬⣍⣓⠤⣄⠻⡦⣿⡜⢣⣿⢴⠟⣠⠤⣚⣩⡵⡲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣳⣦⣀⣀⣈⣳⣙⣄⣉⣻⣛⠿⢮⡷⢾⡵⠿⣛⣟⣉⣠⣋⣞⣁⣀⣀⣴⣞⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡽⠟⠉⠉⢉⡽⣩⠋⣉⣽⣭⣷⢞⡷⢾⡳⣾⣭⣯⣉⠙⢍⢯⡉⠉⠉⠻⢯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠮⢞⣋⡭⠒⠋⣴⠗⣿⢣⡞⣿⠺⣦⠙⠒⢭⣙⡳⠵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡾⠭⠒⣺⢏⠀⢣⣤⠊⠀⡜⡟⢹⡏⢻⢳⠀⠑⣤⡜⠀⡹⣗⠒⠭⢷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡼⢨⢖⡝⠉⠒⡞⣸⡤⢺⡗⢤⣃⢳⠒⠉⢫⡲⣅⢧⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢗⣫⠬⡵⡾⢡⡩⠞⢹⡏⠓⢝⡌⢷⠮⠥⣝⡺⣮⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠟⠒⠁⠀⠀⢰⢣⠏⠀⠀⢸⡇⠀⠀⠹⡜⡄⠀⠀⠈⠒⠻⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⣠⠋⠀⠀⠀⠀⠀⣾⠏⠀⠀⣰⢿⡿⣆⠀⠀⠹⣷⠀⠀⠀⠀⠀⠙⣄⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⡿⠁⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠈⢿⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⢀⡄⠠⡔⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠓⠒⠒⠒⠂⠸⢄⢸⣄⠀⣸⣤⡡⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣃⠀⠀⠀⠀⢀⣀⣮⣀⡉⠛⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⠉⠉⠉⠁⠁⣀⡬⠶⠈⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢐⢲⣾⣿⣿⣥⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠘⣄⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠐⣄⠐⠃⣠⠀⠀⠀⠄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⠀⠉⠐⠛⡄⠠⠟⢲⣂⠀⠤⢄⠉⠉⣠⠔⠊⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⣄⠀⠚⠇⠀⠘⣧⣀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⣠⠴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⣧⠀⠀⣀⡴⣇⢈⡗⢂⠔⠠⠄⢤⠐⠢⠀⠊⠁⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣷⢀⠾⠉⠀⠙⠢⠁⡾⠁⠀⠀⠈⠀⢠⡀⠴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢠⠁⠀⠨⠀⠀⡀⠄⢧⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢦⣀⠄⠀⠀⠀⠀⡼⠀⢸⠆⠈⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣧⡀⣞⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠁⡿⠀⠐⠰⠚⠀⠀⢹⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢹⣠⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣨⠄⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡼⢹⠇⠀⠀⠀⠀⠠⠚⢘⡇⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠉⠉⡀⠚⠃⠸⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠸⠀⡷⠀⠀⡀⠠⢤⡀⠐⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⡀⠀⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢻⠀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠐⠎⠡⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⡴⠞⠁⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠰⢦⣤⣀⣀⣤⠴⠂⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠸⡇⡘⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡄⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠻⣦⡀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠈⠙⠓⠶⠶⠞⠋

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠠⢟⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄
⠀⡼⣤⢻⡏⠱⣾⣶⣶⢩⣶⠞⢢⣤⣀⠺⢢⢀⡀⢸⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣾⣷⣯⡷⠶⢠⡄⣀⣀⠬⡀⣀⣰⢸⡇⢰⡀
⠀⠾⣋⠴⡏⠲⣿⣿⠗⢏⠋⢘⡋⣻⠕⣇⠟⣿⠷⠞⣎⣘⡆⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⣼⢆⣸⡳⡔⢊⡅⣳⡓⣜⢸⣿⡿⡟⠠⡧⠽⣏⡙⢧⣽⣇
⠀⣟⠹⣤⣑⡾⣱⠹⡆⣵⡤⠾⡀⢟⡅⢻⢆⣋⣄⣨⢥⡭⠌⣿⣾⢶⡎⠿⣶⣶⡟⣿⣿⢴⡾⢇⢺⡃⢩⡁⣨⣿⡙⢤⣽⣷⠔⡳⢈⠲⣯⠤⣾⣦⡽
⠀⠑⠀⣾⡹⢆⣯⢀⣿⣇⡓⢛⣀⣋⢱⢾⣯⠔⡰⢣⡜⢻⠳⢼⢷⣸⠗⢂⠸⣿⢏⢹⢇⢚⣝⠺⣿⢳⡏⢻⠀⢇⠑⣤⡈⡛⠁⢤⠘⡄⢼⡐⡗⡸⢧
⠈⠀⠈⠻⠁⠴⣏⠹⠬⠘⠁⠛⠋⡔⢦⠟⣃⠹⢤⠎⠱⡌⢳⠦⡛⠇⠨⠤⠀⠘⠉⠋⠑⠃⠔⠬⠿⠞⠳⣼⡓⠢⣥⠆⡸⠙⡆⣨⠀⢰⢀⠛⡥⢁⠲
⠀⠀⢀⡆⠠⡆⡀⠓⢌⡁⠀⠀⠀⠀⠀⠀⠋⠁⠉⠓⠀⠘⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢷⠀⢦⠈⠈⢶⢌⠛⡛⠒⠀⠀⠀⠀⠈⠉⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠉⠉⠐⠈⠂⠀⠁

⠀⠀⠀⠂⠄⠀⠀⢀⠀⣿⢀⢀⠀⠀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠸⣿⡀⠀⠀⠀⠀⠄⠀⠀⣠⠿⡔⠠⢤⣶⣔
⠀⠐⠂⠀⠀⠠⠀⡀⢱⠇⠉⣠⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢻⣧⠀⠅⠀⠀⠀⣀⣴⠋⠀⠈⢷⠀⢹⠀⡀
⠐⠀⠀⠀⠁⢀⣼⠛⡄⠀⣾⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣇⠀⠁⠀⡠⢚⡁⠀⠀⠐⠀⠑⠪⠀⠠
⢀⠀⠀⠀⢰⡿⠃⠀⠘⠦⡘⢿⣿⣆⠀⠀⠀⠀⠀⢀⠀⠀⣐⣶⣶⣿⣿⣀⠂⠈⠉⠽⠧⣶⡆⣶⣚⠁⠑⠂
⠀⠀⠀⠠⠾⡶⠀⠀⠀⣤⠤⠂⠘⣿⡆⠀⠀⣀⣴⣼⣗⣴⣿⣿⡿⠿⠿⣛⣛⣛⣻⣿⣶⣤⡀⢝⠙⠁⠄⠀⠈
⠀⠄⠘⣷⣙⣁⣶⠀⡔⠫⠂⠀⠀⢸⣿⣠⣼⣿⡿⠿⣛⣩⣥⢶⣾⣛⣿⠽⠿⠿⠟⠛⢻⣿⣷⠘⠏⠀⡀⠂
⠐⠀⠀⠁⠀⣴⣝⡀⣯⠢⢀⣠⣴⣾⡿⢟⣫⣵⢶⣻⡯⠷⠛⠋⠉⠀⠀⣀⣀⣤⠤⠤⣾⣿⡏⠀⠀⠆⠱
⠘⠀⠀⠀⠀⢈⣿⠀⠃⣴⣿⠟⣫⣵⢾⣿⡿⠞⠉⠁⢀⣀⡤⠴⠒⠛⠉⢁⣀⣠⣤⣴⣿⡏⠀⠠⠼⠀⡤⠑
⠀⠀⠀⠛⠉⠁⠸⠀⠀⣿⣏⣼⢫⡿⠋⠁⢀⣠⠴⣚⡩⠅⠀⠀⠀⠀⠈⡉⠁⠀⠀⠀⠹⣿⡄⣀⠗⠀⣈⠳⠐
⠀⠀⠀⠀⠀⡢⠠⡀⣀⠈⠻⣿⣿⣷⣤⣞⣫⡴⠋⠁⣾⣿⠄⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⣿⡇⢐⣀⠠⣄⠋
⠀⠀⠠⣠⡼⠀⡐⣤⠏⢶⣂⠄⠙⠻⣿⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⢀⡤⠀⠀⠀⣿⠇⣿⢿⡇⣡⣤⡀
⠈⠀⠀⢰⡿⣼⡅⠁⠀⠀⠋⢲⠀⠀⣿⡇⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⣼⡟⠀⣿⣿⡿⣋⣽⣿
⠀⠀⠐⡿⠁⠀⠭⡉⠁⣴⣿⠈⠀⠀⢸⣿⡤⣄⡀⠀⠀⠙⢿⣶⣶⣶⡾⢟⣁⡤⠤⠔⣿⣷⡶⠟⢻⣾⠟⡡⠈
⠀⠀⠈⣱⣇⠀⠈⠓⠂⣉⣋⣡⣶⣶⣿⣟⠙⢾⠉⠉⡷⠒⣶⠒⠒⠂⠘⡁⠀⣀⡤⠞⠛⠷⢭⣳⢜⣿⡆⠑
⠀⠀⠂⣾⣣⠀⣠⣾⢿⣻⠿⠟⠋⠉⠁⠈⢳⠞⠀⣸⠃⢀⡿⠤⠤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⠠⠄
⠀⠉⠃⠀⢃⣼⣿⣟⠉⠀⠀⠀⠄⠀⣠⠖⢁⡠⠶⠿⠴⠾⠦⠤⠤⢤⣤⣴⠶⠶⠶⠶⠚⠛⠛⠛⠉⢻⣷⠈⠄
⠀⠈⠀⣴⣿⡟⠉⠹⣧⡀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣧
⠀⠂⠀⢿⣯⡀⠀⠀⠈⢻⣶⣾⣥⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠠⠐⡌⠻⣿⣦⣤⣴⡿⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⡇
⠐⡀⠀⣸⠃⠀⢍⠉⠁⠐⡀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠰⣾⠋⠀⠀⠀⠉⠒⣆⠙⡀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⣿⡀
⠀⠀⠀⡘⠛⠻⠿⣿⣿⣤⣾⣽⣿⢏⣤⠖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠀⠀⠨⣿⣿⣿⣆
⠀⠀⠀⠙⠻⠷⣦⣶⢟⣿⠿⠛⠋⠉⠀⠀⠀⠀⠠⢉⣍⣁⡒⠛⣀⣘⣂⡀⠀⠀⠀⠀⠀⠘⢼⡿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠒⠄⠀⠐⠶⠤⠀⠀⣀⠒⠚⠂⠀⠬⠉⢛⣀⠀⠀⠀⢈⡖⣫⣎
⠀⠀⠀⢠⡞⠉⠀⠀⢽⣿⣿⣿⣷⡄⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢇
⠀⠀⠀⠘⡂⠀⠀⡀⠛⠋⠙⠋⣉⣇⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣾⢿⣆
⠀⠀⠀⠸⣿⣧⣴⣶⣷⣶⣿⣿⡿⠷⠀⢀⣀⣀⡀⢬⠐⢌⠀⢅⠠⣍⠑⢌⠁⣤⠳⠀⠀⣀⣀⠀⠀⠀⢹⠛
⠀⠀⠀⠀⠙⠯⣶⣦⣾⣿⣽⠿⠋⠀⠀⠀⠀⠀⠐⠂⠐⠄⠀⠀⠀⢀⡀⠀⠀⠈⠀⠬⠀⠋⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⣆⠻⢷⣆⡈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠐⠒⠒⠦⠀⠀⠀⠀⠠⡴⠖⢒⡂⠤
⠀⠀⠀⠘⣿⣦⡈⠻⢿⣦⣄⡄⠉⠲⢤⡀⠀⠀⢀⣾⠟⠁⠠⣄⡠⠾⢷⣷⡄⠀⠀⠀⢸⡇⠀⠈⢻⣿⣶⠄
⠀⠀⠀⠀⠘⢿⣿⣦⣄⠉⠛⠿⣿⣷⣈⣻⠀⠀⠼⣿⣄⣀⡄⢈⣦⠀⠀⣽⡆⠀⠀⠀⠀⢻⡄⠀⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣶⣶⣬⣿⠛⠀⠀⠘⢮⣏⣩⠄⠸⢋⣠⡼⠋⠀⠀⠀⠀⠀⠐⠧⢈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⡠⠀⠀⠑⠀⠀⠀⠘⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⡈⠐⠀⠀⠀⡀⠀⠀⢀⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⢰⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠢⢄⠀⠁⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⠠⠊⠀⠀⠀⠀⠑⠀⠒⠀⡀⠀⠀⠐⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠑⢒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⡀⠀⠸⠀⠀⠀⣀⠰⠀⠈⠀⠀⠀⡇⠀⠀⠀⠙⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠱⡁⠉⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⢠⠀⢀⡉⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢀⢀⠀⠀⠀⣀⠀⢀⠀⠣⡈⠓⠢⠤⠤⠤⢐⡙⠀⠒⠉⢀⡌⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠈⠰⠁⠀⠀⠈⠁⠀⠂⠒⠚⡉⠀⡑⢲⠒⠍⠀⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡀⢈⠢⠄⣀⣀⣰⠀⠀⠀⠈⠣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⢰⡀⢀⣀⡀⠀⣄⡀⠠⡄⠈⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⠢⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠴⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠂⠀⠀⡀⠀⠂⠀⠀⢄⡴⣤⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⣿⣓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠦⠤⢄⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠠⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠐⠀⠐⠵⠶⢄⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠈⢀⡀⠛⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢔⡠⠆⡠⣀⢄⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠞⢁⡄⠏⢠⢄⢤⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡆⣠⢠⡀⣉⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢰⠇⠀⠠⠸⠟⠽⣈⡀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣱⢀⡁⢍⣬⣙⢢
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠽⠡⠠⣤⡀⡀⠀⢀⢀⠀⠀⠑⡴⠊⠐⠔⠂⠀⣷⣭⡥⠄⡛⠚⠿⠻⠤⠐⠂⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⣀⣒⠓⠧⠗⠀⠾⠿⡇⠀⡧⣿⣏⣷⠁⠈⠃⠀⠿⠓⠚⣇⢀⠠⠀⠦⠔⠐⠒⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠸⠒⠀⠀⠐⠒⠸⠔⠒⠢⣼⠀⠐⡆⡠⠴⠊⠋⠉⠉⠐⠀⠀⠀⠓⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠺⣄⠤⢄⣀⣀⠀⠀⣀⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠴⢠⣼⣿⠿⢛⣽⣿⣿⠟⣿⣿⡿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠀⢈⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⡛⣿⣿⣷⣤⣁⠛⠿⠤
⠀⠀⠀⠀⣦⣴⡶⠒⠀⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣼⣿⣯⣿⣿⣿⣿⣌⣛⠻⠟⣶⣶⣤
⠀⠀⠀⠀⣀⣠⡴⠋⣽⣿⣿⣿⣟⣻⣿⣿⣿⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣮⡛⠳⢶⣶⠖
⠀⠀⠀⠈⠉⣁⠀⢺⣿⣏⠻⣿⡙⢿⣿⣻⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣠⣤⣿⣿⣿⡿⣿⣿⣿⣏⣿⡿⢻⣿⣿⣿⣛⣿⣿⣿⣷⡦⣬⡤
⠀⠀⠀⠰⠟⢣⣶⡚⠿⣿⡆⠈⠓⠈⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⡿⠁⠀⣴⣯⡽⢿⣿⣯⣿⣿⣿⡿⣷⣗⡀
⠀⠀⠀⠀⢠⢠⣞⠻⢦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣤⣀⣀⡀⠀⠀⠀⣀⣀⣿⣿⣿⣿⡟⠀⠀⠈⠋⢁⣴⣿⣿⣶⣿⣯⣿⣷⡌⠋⡈
⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠘⣉⣩⣾⢻⣿⣆⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣹⣾⡿⢉⣾⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⣀⠀⠉⠉⢁⡀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣿⣿⣿⣽⣿⣿⣿⣿⢳⡄⠙⠻⠛
⠀⠀⠀⠀⣠⠄⣩⢴⣽⣿⠿⢡⣿⢿⣿⣿⣿⡇⠁⠀⠀⠀⠉⢀⡀⠀⠘⠃⠀⠀⠈⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠱⣶⣠⣀
⠀⠀⠀⠀⣤⣾⡿⠟⠋⢡⡖⣿⣿⣿⣿⣿⣿⣧⢃⠀⠀⠖⡀⠻⠿⠦⠄⠰⠆⠀⠰⣸⣿⣿⣿⣽⣿⣿⣿⣭⣿⣿⣿⢿⣿⣿⣷⡈⠻⡶
⠀⠀⠀⠀⠉⢁⣴⣿⡾⢂⣺⣿⣿⣾⣿⣿⣷⣾⣧⣠⡀⠠⠇⡃⢲⢰⡎⠃⠀⢀⣵⣿⣟⣹⣿⣿⣿⢻⣿⣏⣿⣿⣿⣆⢭⣛⡛⠿⠦
⠀⠀⠀⠀⠚⠋⠛⢋⣴⣿⣿⣯⣿⣿⣽⣿⢿⣿⣿⣿⢻⣶⣤⣥⣍⣀⣤⣴⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣉⣉⠛⠗
⠀⠀⠀⠀⣠⢴⣎⣻⣿⣿⣿⠿⣿⣿⣿⣿⣾⣿⠟⠙⠿⢿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⢻⣿⣿⣿⠿⣿⣿⣿⢦⣿⣧⡈⣉⣡⡌⠛⠷⡄
⠀⠀⠀⠀⣺⣯⣿⠟⣋⣼⣿⣿⣿⣻⣿⣿⠋⠁⠀⠀⠀⠀⠈⣿⣽⣿⠁⠀⠀⠀⠀⠉⢿⣿⣷⢿⣿⣿⡯⣭⣽⣿⣿⣿⣦⣭⣈⣛⣢
⠀⠀⠀⠰⠊⢉⣀⣾⡿⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⠇⢁⠀⠸⣿⣿⣿⣝⠷⠷⣟
⠀⠀⠀⠈⣱⡾⠛⣩⣾⣿⣿⣿⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣡⣀⣸⣄⣀⣈⣿⢿⣿⣇⣠⣍⣀
⠀⠀⠀⢀⣥⣴⣾⡟⠉⠓⣩⣾⣿⣿⣿⠏⠳⠤⣀⡀⠀⠀⠀⠈⣿⣿⣿⡄⠀⢀⡠⠴⠋⢻⡭⠭⡇⠀⠀⢸⠁⠀⠀⣿⠆⠛⢿⣿⣻⡆
⠀⠀⠀⢠⣴⣶⢶⡶⠂⣶⣿⣼⣿⣿⣿⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣽⡿⠃⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⢸⠀⠀⠀⣿⣗⡾⢶⣬⣉⠃
⠀⠀⠀⠰⠟⠋⣀⡴⣿⣭⣉⡉⠛⠻⢿⡀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⡇⠀⠀⢸⠀⠀⠀⣿⣿⣿⣬⡙⠻⠿
⠀⠀⠀⠀⠶⣼⣿⣻⣿⠿⠿⠿⢿⣶⣼⠳⣄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣶⡇⠀⠀⢸⠀⠀⠀⣿⣿⣿⡾⣿⣢⣄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⡀⢀⣼⣿⠁⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠁⠀⠹⣟⠃⠀⠀⠸⠦⠐⠐⠉⠉⠛⠛⠻⠶
⠀⠀⠀⠀⢴⠶⠖⠓⢂⣀⣠⣽⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡀⠀⠀⠀⠤⢀⣀
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣶⣤⣤⣄⣉⣉⠐⠒⠒
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣶⡶⣯⣽⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢿⠿⠿⡿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡛⢋⢻⣛⡚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠉⠈⠢⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠒⢤⡀⢸⡁⠛⠀⢀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⠉⠋⠀⠀⠀⠀⠀⠊⠂⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠐⠄⠚⠒⠔⠰⣶⣶⣶⣶⣶⣾⣷⣦⡀⠀⠀⡆⣀⣸⣿⣿⣛⣏⣻⣿⣿⠆
⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⣿⣿⣿⣿⣿⣿⣿⠟⠂⠐⢿⠛⠻⢿⣷⣬⣦⡟⣿⠃
⠀⠀⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⢋⣠⠆⠀⠀⠀⢡⠀⠀⠈⠙⠛⠻⠿⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢁⠂⢘⣃⠀⠀⢘⠀⠀⠀⠀⠤⣦⣷⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠉⠀⠀⠈⡩⠂⠈⢢⡈⠀⢿⢷⣶⣶⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠀⠒⠁⠀⠀⠀⠀⠑⣠⣿⣿⣿⣿⡿⠣⠀⢰⣤⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⢛⠟⡟⠁⠀⠀⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⡂⠀⠀⠀⠀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⢀⡀⠅⠊⠉⠀⠰⠗⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠟⠛⠋⠉⠉⠉⠙⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠉⠛⠛⢲⣾⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠈⠉⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠤⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣿
⠀⠀⣿⣇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣾⠏
⠀⠀⠘⢿⣦⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠄⡀⠂⡀⢂⣥⡾⠋
⠀⠀⠀⠀⠙⢷⣦⣥⣈⣀⠄⠀⢀⣰⣤⡶⠏⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠾⣤⣥⣀⣤⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠙⠲⣥⣬⣈⣠⣂⡄⣐⣀⢀⣅⣈⣤⡬⠿⢷⣬⣄⢄⠀⠄⠤⢀⢠⠠⣠⣴⠟⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠈⠙⠛⠳⠿⠶⠶⠛⠛⠉⠀⠀⠀⡀⢀⠀⠀⢄⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⣣⠆⠀⠪⠃⠊⠠⡆
⠀⠀⠀⠀⠀⠀⢈⣐⡸⠀⠙⠋⠈⢀⢩⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠁⢬⠳⠀⢀⠠⠐⠕⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⢂⡐⠀⠤⡀⣱⠩⡈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠒⢌⠂⠐⡀⠭⠌⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡔⠮⠂⠄⠌⠔⠀⠙⠈⠃⠀⠀⠀⠀⠀⠀⢀⢐⠔⡎⠄⠒⠠⢑⢄⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠡⢰⠢⠀⠂⠖⠕⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠂⠄⡬⠆⡠⡆⢌⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⠀⡤⠀⠀⡞⡕⢂
⠀⠀⠀⠀⠀⠀⠈⠤⠁⠠⣦⠦⠀⢁⠅⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡨⢲⠁⢠⣤⠀⠰⡰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠂⠂⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡏⠑⠂⠀⠀⠀⠀⠀⠀⡀⡐⠀⠀⠀⠀⠀⠀⠁⠀⠊⠰⠐⠩⠒⠂⠀⠀⠀⠀⠀⡉⠠⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡆⠐⠭⠊⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄⠑⠨⠃⠀⠀⢼⠡⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⢀⢠⠀⡀⢄⢒⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠲⣁⠄⠄⠒⡕⠀⢁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣒⡄⠀⣡⢀⡒⠀⢀⣁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⠘⠠⠰⠂⠠⠩⡎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢂⠜⢀⣀⣄⠀⠐⣑⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣒⡆⠀⠤⡔⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢚⠢⠤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠚⢈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠖⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠛⠀⠉⠉⠁⡇⠸⠈⠉⠁⠀⠁⠛⠀⡇⠀⠀⠀⠀⠀⣠⠈⡤⠤⠠⠄⠀⠀⠤⠤⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⡇⠐⠀⠀⠄⠠⣤⣤⠀⡇⠀⠀⠀⠀⠘⠀⡀⠀⠄⠀⠒⠒⠒⠒⠒⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⢠⡀⢰⣔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠛⠚⠂⡇⠈⠘⠒⠃⠈⠛⠃⠀⡇⠀⠀⠀⢀⠀⡆⠈⠋⠈⠉⠀⠀⠀⠀⠀⠀⡘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⡇⢀⠀⡀⡀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠇⠈⠦⣀⠒⠘⠋⠁⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠈⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠈⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠒⠒⠂⡇⠐⠐⠒⠂⠀⠺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣠⣤⣶⢤⡗⠊⠛⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡵⡖⡄⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠀⡇⠈⠀⠀⠀⠘⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠌⢳⡏⠸⠈⠂⠇⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠀⠤⠤⠀⡇⠀⠠⠤⠄⠠⣤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢷⠘⡀⠀⠘⢶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⢳⡇⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠙⠛⠁⡇⠸⠘⠋⠃⠐⠓⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣠⣠⠀⢀⠀⠀⡇⢠⠀⠀⠀⢠⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⠆⠀⠒⠒⠂⡇⠐⠐⠒⠒⠰⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⢀⠀⡇⢈⠀⣀⡀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠀⠠⠤⠀⣷⡆⠠⠤⠆⠀⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠀⠀⠉⠀⣿⡇⠀⠀⠁⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⢠⡶⣿⣿⣶⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⢿⠿⠿⠿⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠀⠀⠤⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⡄⠤⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⠀⣀⢌⣳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣀⠴⡁⠈⠀⠠⢗⠎⠱⣄⢀⡤⠤⠔⠒⠦⢴⣁⠀⣈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠳⣤⡀⠀⠀⠈⢉⣹⡃⠀⠀⠠⡈⠄⡀⢌⠋⠀⠀⠀⢀⠀⢴⠖⠈⣠⢙⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠂⠀⠈⠻⣅⣀⠠⠶⢭⡤⠃⢀⡠⠒⣐⠖⠀⣤⠀⢀⡐⠀⠁⠈⠐⡀⠀⠠⡌⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢫⡅⠂⠀⠈⠁⢶⣉⣠⡖⠋⠀⠀⣀⣀⣐⢁⠀⠀⣈⠳⢼⣅⡀⠀⢀⣰⠇⠀⢀⡈⠯⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡴⠀⠀⠀⠰⠉⠁⠀⠡⠐⠂⢄⠀⡈⠙⡁⣀⡀⠈⠁⣻⣾⣉⣉⡅⣬⢧⣀⡄⠀⡒⢶⡏⠤⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠁⡇⡠⠊⠀⠀⠄⢀⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣇⠀⠀⢄
⠀⠀⠀⠀⠠⠒⠀⠀⠀⢂⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⢺⠀⠄⠁⠬⡅⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢹⢰⡂
⠀⠀⠀⠀⠀⠈⣒⠀⠀⠀⢣⠠⠤⠤⠤⠐⠰⠂⠀⠚⠀⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⢀⣀⡤⡖⠁⠀⠀⠀⢜⡚⠈⠁⠁
⠀⠀⠀⢀⠀⠀⠑⣄⠀⠐⠀⠀⠀⠃⠀⠀⠐⡀⠀⠀⠀⠀⠏⣀⠀⡀⠉⠤⠀⠀⢠⠏⠀⠀⠛⡄⠀⠀⠈⠀⠈⠠
⠀⠀⠀⡆⢀⠐⢄⡤⠤⠀⠀⠀⠀⠰⠰⠀⠀⠀⠢⢄⣀⠤⣄⠁⡩⠈⠓⠈⠀⣠⠃⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢲⠀⢰⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿⠉⠀⣠⣀⠠⠚⠁⠀⠀⠀⢰⠀⠀⠀⠠⡀⠀⠀⢀⠃⠀⠀⠀⡀
⠀⠀⢀⣀⠀⠀⠀⠀⠠⢸⣀⠜⠂⢒⣀⡀⣀⣀⣸⣀⠐⠒⠻⣚⠿⠍⠀⠀⠐⠀⢀⢂⣀⣈⢅⠐⢲⢴⡹⠡⣄⣀⡁⡱
⠀⠀⠀⠂⠐⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⢤⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠀⠀⠀⠀⣀⣹⡄⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⡴⠛⠉⠁⠀⣠⣷⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⢰⢶⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣄⣀⣀⠀⣀⣀⡴⠋⠉⠙⠋⠉⠀⢀⣠⣴⣾⣿⠿⠓⠚⢉⣩⠭⠟⠁⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠁⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣷⠈⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠉⠀⣠⠴⣚⠉⠀⠀⠀⠀⣰⡄⠀⠀⣿⠀⠀⢸⢸⠀⠂⠠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⣀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠛⠁⢀⣤⠖⠋⠁⠸⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⡿⠿⢛⡹⠛⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⢀⡀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠛⠁⢁⡤⢒⣉⡤⠖⠋⠁⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⡀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣀⣀⣀⣴⠾⠟⠚⠉⠀⠀⠀⣠⠶⠶⣄⠀⢿⠇⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣤⡿⠉⢉⡇⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢧⠀⣀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⣟⠁⣀⣀⠀⠀⢹⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⡈⣳⡘⠊⠃⠀⠀⠀⠀⠛⢦⣄⣴⠇⠀⠀⠀⠀⣦⣠⠤⡴⠋⠀⠀⠸⡄⠀⢸⠀⠀⢸⢠⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣾⠉⣉⣙⣓⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⡀⠀⠀⠀⠀⢬⠀⢸⠀⠀⢸⣤⠀⠁⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⣿⣿⣿⣿⣿⣷⣾⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠱⡄⠀⠀⢀⠜⠀⢸⠀⠀⢸⣿⠀⠂⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⡄⠀⠀⠀⠰⣶⠞⣡⠟⠀⠀⠀⠙⢠⡘⡯⠀⠀⢸⠀⠀⣾⣿⣷⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡓⠀⠀⣀⡴⠛⠋⠁⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⢸⠀⠀⠙⠛⠁⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢾⠁⢤⡀⠀⠀⠀⠘⣆⠀⠀⠸⠄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⢳⣠⠛⣆⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠠⣶⣄⣿
⠀⠀⠀⠀⣴⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡠⢻⡆⠀⠀⠀⠘⣦⠀⠀⠠⠶⠀⠀⠿⠾⣧⣦⣴⣿⠁⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠻⣿⣿⣿⣿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠃⣾⠆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠓⢦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⠀⠀⠈⠙⢦⠀⠀⢿⢳⣄⠀⠀⠀⠛⠋⠀⠀⠀⠀⢀⣿⠀⠀⣿⠇
⠀⠀⠀⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⠇⠀⢸⡆⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠉⠀⠀⠙⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⠆⡔⡠⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠾⢾⡴⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠙⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠻⠟⠛⠿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠠⣾⣾⣿⣶⣄⠀⠀⠀⠀⠀⠘⢷⣌⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⢀⠸⢿⢙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣧⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢤⡟⢸⣃⣬⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢠⡶⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡄⠀⠀⣉⣦⡆⣿⣾⠓⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣴⣶⣫⣅⣀⠀⢾⡇⣽⢾⣦⣻⢿⣞⣿⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⢛⣼⣋⢠⡀⢰⣛⣸⡇⣉⢋⣹⣻⣿⣍⣭⣦⡀
⠀⠀⠀⠀⠀⣄⠀⡀⠘⠖⣿⢠⡀⠐⠂⣀⡀⠀⠀⣀⡄⣂⡾⣻⢷⢋⣿⣿⣻⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠄
⠀⠀⠀⠀⠀⢬⡻⢤⠤⢼⠿⣏⡀⣤⡄⠀⠀⠐⠂⠉⢙⣋⡶⣉⡄⠙⣿⣿⡟⠿⢟⣫⡿⢟⡿⣿⣿⣿⣿⣋⣥⢼⣇⣶⣄
⠀⠀⠀⠀⠀⠀⠠⠈⣤⣤⡤⣿⣧⡄⠳⢄⣤⣤⣀⠲⣾⣿⣴⠏⢠⠟⠹⣿⣤⣦⣟⣉⣷⢟⣿⡟⣿⣷⣾⣿⢻⣿⡿⠿⡝⣄
⠀⠀⠀⠀⠀⠀⠀⢾⣄⠐⠂⠞⢻⣧⣀⢀⣍⠛⢋⣴⣿⣛⣥⡆⠀⣠⣾⠛⢻⣟⣿⣿⣷⢬⣿⣼⣿⣿⣿⢿⡼⣿⡇⣶⣷⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⢄⣠⠼⢿⣿⡿⢃⣥⣿⣁⡿⢧⣿⣀⡾⢻⣇⣵⠟⢉⣽⣿⢿⡿⣿⣿⣿⣾⢛⣼⣇⣻⣿⣿⣧⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣮⣳⣶⣊⣿⠿⣿⣾⣯⣿⣟⣷⢿⣯⢥⣿⡟⣻⣆⢸⣯⣴⣎⣾⣯⣿⣻⡿⠿⣿⡛⣿⣣⣼⣽⣷⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣭⣿⠟⣽⣾⡟⣻⡿⠿⢿⣣⡾⣻⡾⢻⣿⡿⣿⣼⣯⡿⢿⣿⠏⣹⡿⣷⠿⣿⡿⢿⣿⣿⡿⠻⣾⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣾⣋⣷⣿⣍⣛⠻⣻⣿⣿⢛⣿⣿⣧⣴⣋⣿⣏⣴⢦⢶⣾⣻⣶⣏⣤⣾⣵⣶⣿⣶⣶⣾⣿⣿⣿⣯⡛⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣟⣻⢶⣿⣿⣹⣷⣾⣏⢛⣿⣿⣩⣿⣿⣻⣿⣽⣯⣶⣿⠛⡿⡟⣿⣿⡿⢿⣿⢿⣏⢉⣿⣽⣿⣿⣿⣿⣽⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⡽⣿⣿⣯⠵⣿⣿⣽⣿⣿⣏⣴⣿⣿⣿⣿⣏⣴⣤⣧⡆⢩⣼⣀⠀⣼⢛⢿⠻⢿⡛⠛⣍⢿⣿⣷⡿⣯⠛⠩⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠱⠶⢻⣷⡺⣿⡋⢷⣅⣭⣿⣿⣿⣿⣿⣿⣿⠛⢿⡛⠿⣿⡟⣿⣿⡾⢿⣿⣴⣿⣤⣿⣿⣿⣼⣿⣿⣷⣄⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⣿⣾⣿⣮⣿⣽⣿⣿⠿⣟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣻⣿⣎⣻⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣛⣯⣟⣹⣷⣿⣷⣶⣛⣿⡼⢿⣵⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣯⣷⣿⡿⣿⣿⣿⣿⣿⣿⣯⡢⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⣯⣍⣬⣯⣙⣿⣿⣋⣶⠾⢿⣿⠟⣫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣭⣿⣟⣿⣾⣷⣤⠶⠶⢿⡾⢫⡜⢻⣿⣿⣯⡀⠙⠻⢿⡿⠿⣿⣿⣷⣿⡿⡏⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⡿⠋⠙⢿⣯⣷⣶⠞⣻⡅⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⢇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⣾⣿⡟⠀⠀⠈⢀⣀⣀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢻⡏⠙⠃⠀⠀⠀⣼⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣻⣷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢺⡛⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢲⣟⠛⠙⠋⠛⠛⠉⠉⠀⢀⠀⠁⠐⡇⠀⠀⠀⢠⣤⢠⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡴⠖⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⢀⣶⡀⠀⠀⢀⡇⠀⠀⠀⢸⣿⡜⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡜⢀⡀⠀⣶⠀⢠⣇⠀⠀⣸⠹⣧⠀⠀⣸⣷⠀⠀⠀⡞⣿⣧⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⣠⡄⠀⣸⣇⠀⢠⢿⣷⢠⡇⢸⣇⠀⣿⣆⢸⣿⡄⠀⡅⠀⣿⡇⢀⠅⣿⠀⠀⠀⡇⣿⣿⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣧⡀⡿⣿⡀⣼⠈⣿⣼⣷⣼⣿⢸⢻⣿⣾⣿⣇⠀⡇⠀⡿⣿⢸⠀⣿⡇⠀⢰⠃⢻⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⠀⡇⢻⣷⡏⠀⢻⣿⡿⠛⣿⡿⠀⢻⠁⣿⣿⢀⠇⠀⡇⠹⠟⠀⢹⡇⠀⢸⠀⠀⠀⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⡇⠘⢷⡇⠀⠀⠿⡇⠀⠙⡇⠀⢸⡆⣿⠹⣾⠀⠀⡇⠀⠀⠀⣼⡇⠀⣼⠀⠀⢀⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡏⣿⠋⣷⠀⠀⠁⠀⣰⠀⡇⠀⠀⡇⠀⢸⠇⠻⠰⢹⠀⠀⠁⠀⠀⢰⣿⡇⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠸⡀⣸⡆⠀⠀⠀⣿⠀⠟⠀⡆⢹⠀⢸⠀⠀⠀⣼⡇⠀⠀⠀⢀⣿⢻⡇⠀⣿⠀⠀⡌⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄⠀⣿⣿⠀⠀⠈⣿⡆⠀⠀⠀⣿⡆⢸⠀⠀⠀⣿⣧⠀⠀⠀⣼⢹⢸⣿⢀⢿⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⣿⢹⡇⠀⠀⣿⡇⠀⠀⡀⣿⣇⣸⡄⠀⠀⣻⣿⠀⠀⠀⡇⢸⠸⡏⢸⠘⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⡘⣿⠀⠘⡟⣿⡀⠀⠀⡏⢹⣿⡇⠀⠀⡏⣿⠀⠀⢰⠃⢸⢰⣷⡜⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡏⣿⢿⠀⢰⡷⣿⠀⠀⢠⡇⣼⢹⡇⠀⠀⡇⣿⠀⠀⢸⠀⢸⣸⠻⠇⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⢸⡆⢸⣟⢻⠀⠀⢸⣇⣿⠸⡇⠀⢸⠁⣿⠀⠀⡏⠀⢸⣿⠀⠀⠀⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⢸⣾⠄⠀⢸⢿⠟⡀⡧⠀⢸⠀⣿⡄⢠⠇⠀⠈⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⢿⡇⠀⡾⠈⠀⠃⣟⠀⢸⠀⣿⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⣿⠀⢸⡄⡇⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⢹⠀⣿⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠘⣦⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡎⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢄⡨⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠢⠔⠚⠑⠰⢁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⡀⠒⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢂⠀⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢄⡀⠈⠠⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠂⠉⠉⠉⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⢀⣀⠙⠢⣄⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⠀⠀⣀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡤⠜⠁⠀⠐⠁⠨⡇⠀⠀⠱⢀⠀⠑⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠤⠄⡀⠈⠋⠉⠉⠐⠁⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⣀⠀⠀⠈⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠁⠀⠀⣠⠄⠀⡀⠀⠀⠀⠀⠀⠂⢀⠀⠐⠒⢀⠀⢀⠐⠛⠉⠙⡃⠶⠔⠛⠉⠀⠀⢉⣤⣴⣦⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠁⠀⠀⠀⠀⠈⠈⠀⠐⠁⠅⠀⠀⠀⠀⠑⠺⠄⡛⠀⢙⠛⠀⡀⠀⠒⠀⠐⠋⠁⠀⠉⠛⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠⠀⣀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠁⢘⠏⠉⠀⠀⣛⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⡠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⢀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠠⢠⣤⠀⣀⡀⠀⣠⠴⠀⠀⡉⠻⣀⠯⠤⠶
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠃⠀⠘⠤⠧⠽⠀⠀⡁⠣⡀⠀⡁⠜⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠃⠀⠀⠀⢰⡀⠀⠀⠀⠖⠒⠒⡀⠀⢀⡐⠀⠈⠑⢂⠂⠁⣸⠷⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠠⠄⡀⠢⠅⠀⠀⠀⢀⠤⠀⠘⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠚⠁⠀⠑⠬⢀⡀⠀⠀⢠⠃⠀⣀⣤⣒⡉⠥⠀⠐⠸⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠋⠙⡗⠛⠥⣌⠀⠄⣸⠃⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠈⠠⠀⠀⠑⠢⡉⠢⣄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠴⠟⠂⠀⠈⠑⠠⠈⣈⠁⠉⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⡀⠄⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣄⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠄⠀⠀⣿⠈⢧⡴⣃⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠀⠀⣿⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⣿⠀⠀⠘⢧⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⢻⡆⠈⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⢸⡇⢾⠀⠀⠀⢠⣠⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡞⠀⠀⢳⡌⢳⡄⠲⣄⢹⡇⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠀⠀⠀⠀⢹⠀⣷⠤⢬⣿⣿⠥⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢳⣄⠀⠀⣀⡞⢠⡏⣠⠞⢹⡏⠳⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⣉⣉⣡⡴⠋⠀⠁⠠⠿⠳⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠒⠒⠂⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠉⠐⠒⠂⠀⢴⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⢛⠀⠈⠑⢦⣄⣀⠀⣤⠶⢿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠇⢀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⣌⡐⠈⠲⢦⣤⣤⣠⣤⣴⡫⠥⠒⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠒⠀⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣾⡆⠀⢀⣴⣤⠀⢀⣶⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠘⣿⠟⠁⠈⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⢰⠇⢷⡀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠠⠔⠉⠐⠒⠚⡆⠘⡇⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠛⠒⣤⣀⡀⠀⠀⠀⠀⣳⣶⠗⠚⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡁⢯⠀⠰⢆⠀⢸⡈⠉⠘⡟⠉⠻⠀⢰⡀⠀⠀⣰⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⢆⡡⠀⠋⡀⠀⡗⢲⠒⣇⠀⠀⠀⠉⠻⠛⠈⠀⠀⠀⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠳⠀⠈⠷⠀⠀⡇⠀⡇⠀⡄⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⣦⡀⠀⠀⢸⡁⠀⡇⠀⠇⢿⣤⣤⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠁⠹⣄⣀⣼⡇⠠⡇⠀⠷⣤⣀⣉⡉⠙⠻⠋⠁⠀⠀⠀⡰
⠀⠎⠀⠀⠀⠀⠀⠠⠆⠀⢠⡶⠋⠁⠀⠀⠀⠀⢰⡄⠀⠈⠉⠉⠀⠘⠀⢀⣀⣀⣤⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⣴⠋
⠀⠀⠙⠶⣤⣄⣀⣀⣤⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣠⠾⠿⠻⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠖⠛⠁⠀⠀⠀⠃⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢤⡀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣠⡴⠶⠛⠛⠛⠓⠲⢤⡀⠀⠀⠀⣠⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠀⠀⠀⠀⠈⠀⣠⡼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⡄⢣⢠⠁⠀⠀⠀⠀⠀⠠⢬⡄⣰⣾⠁⠀⠀⠀⠀⠀⠀⡄⠀⡨
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠋⠀⠠⠀⠀⠀⠀⡂⢤⡄⢄⡦⢄⠀⠀⠀⠀⠈⠀⠘⣔⣅⠤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠬⠭⠕⢘⢛⢻⠅⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢃⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠂⠴⢶⡋⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⡠⠌⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⠔⢏⠀⠀⠈⡼⠑⡜⠀⢀⡠⠊⠁⠀⠀⠀⠀⠁⠀⠱⠚⠒⠎⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠢⠀⠀⠐⠀⠀⠠⢀⡡⠀⠈⡇⠀⠉⠐⠉⠀⣴⠊⠁⠀⠁⠀⢄⣀⢀⢀⡂⠐⠶⠚⠲
⠀⠀⠀⠀⠀⢴⣤⣨⡦⠐⠶⠜⠶⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣈⣉⢁⣉⡑⠶⠶⠴⣴⡄
⠀⠀⠀⠀⠠⣖⢤⠤⣴⠲⠶⠻⠔⠴⠄⠀⠐⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⠈⢉⠉⠉⣶⣄⢠⣤⣄⠴
⠀⠀⠀⠀⢐⣠⣀⣤⣬⢖⢰⣠⢤⠠⣄⠐⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⡋⠉⠉⠉⠙⠮⣌⠠⡀⢈⠁⠀⠀⠄
⠀⠀⠀⠀⢄⣠⡠⣄⣬⢅⡀⢀⠀⡀⣀⡠⠀⠀⠈⠂⢄⠈⠁⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠛⠓⠓⠘⠓⠓⠦⣿⡆
⠀⠀⠀⠄⠄⢀⣀⢀⠍⡉⠛⠙⠛⠉⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⡆⠀⠀⠀⡾⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⣇⠀⠈⠠⠁⢺⡆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠠⠀⠀⡁⠀⠀⠀⠀⡌⠉
⠀⠀⠀⠀⠀⠠⣁⠀⠀⠀⠤⠆⡗⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠬⠀⠀⠀⠀⠉⠤
⠀⠀⠀⠀⠀⠀⠉⠓⠶⠀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠁⠒⠁⠛⠀⠀⠀⠐⠀⠠⠀⠀⠈⠀⠀⠐⠂

⠀⠀⠀⠀⠀⣀⡠⠴⠒⠒⠶⠖⠒⢤⠄⠐⠒⠒⠒⠂⠀⠀⠐⠒⣠⣄⢀⡤⠖⠒⠒⠒⠒⠒⠒⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠻⠤⡀⣀⡀⣐⠤⠖⠚⠛⠋⠀⠀⠠⣀⡀⢀⡀⠀⠀⠀⠈⠉⠉⠉⠩⠽⢯⣄⣀⣀⣀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢃⡏⢙⡧⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⣻⣦⠶⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠦⢤⠀⣠⣄⠀⢀⠀⠀⠀⠑⠦⢤⣄⢠⣶⠶⣸⢟⠀⠀⢲⢸⣿⡞⡏⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⢏⠘⡇⡟⠀⠀⠀⠀⠀⠀⠹⢸⡇⠀⣏⣌⠀⢠⡿⣼⣿⡇⣁⣸⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣷⡆⢠⠖⠀⠀⠀⠀⣾⡁⠀⣼⣿⠀⢸⣇⢻⢸⠀⢻⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡏⣿⡇⢸⡆⠀⢰⡄⠀⣿⡇⠀⡿⢹⠀⣿⣿⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣷⢸⣿⠀⣼⡇⢠⡇⡇⢰⡇⢸⢘⣿⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣲⢿⢸⢸⣿⠀⣿⠇⢸⡇⡇⣸⡇⣷⢸⣿⣿⡀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⢸⣿⡇⣿⢰⢸⣧⡇⠏⡇⣇⢸⣿⣿⡇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⢸⡇⡇⣿⢸⠸⣷⡷⡆⡃⡟⢸⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣇⢹⣼⠀⣿⣿⡄⠀⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇⣿⣾⣿⡆⣿⣿⡇⠀⡿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇⣿⢻⢿⣧⣿⣿⡇⠰⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⣼⣿⠷⠿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠛⢻⠀⠀⣿⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠾⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⡤⠶⠶⠶⡶⠛⠛⠛⠻⠶⠛⠛⠛⠶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢳⣄⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣾⡏⠀⡤⠖⠛⠛⠓⠀⠀⠀⠀⠈⠉⠉⠉⠙⠲⠄⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠘⠶⠶⠶⠚⠁⠀⠀⠀⠀⠘⠓⠲⠒⠛⠀⠘⣧⠀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⡇⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣠⡤⠀⠀⠀⠀⠀⢠⣿⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣬⣿⣄⣀⣀⣀⡀⠀⠀⠀⠉⣉⣁⣀⣀⣀⣠⣤⣴⡿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠉⢙⣷⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠁⠀⠀⠈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⠀⣠⡴⠟⠋⠁⠀⣦⣤⡾⠿⢿⡟⠷⢶⣶⣶⣶⣿⣿⡟⢛⣛⣩⣥⣤⣴⠶⠶⣤⣄⡀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⡴⠟⢛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⣆⠀⠈⠉⠛⠛⠉⠁⠀⠸⣇⠀⠀⠀⠈⠙⠳⣦⡀⠹⣿⡄
⠀⠀⠀⠀⣾⣣⣴⠿⢻⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⡀
⠀⠀⢀⣼⡿⠋⠁⠀⠸⣇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣄⣀⣀⣼⡃⣿⣇
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⣆⠀⠈⢷⣄⡀⠀⣀⣀⣀⣀⣀⣀⣀⣼⣧⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢷⣼⣿
⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢈⣿⡏⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣀⠀⠀⡰⠾⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢬⢺⣁⠀⠀⣰⠀⠀⠀⣂⢠⣭⢀⣶⣠
⣤⣤⢧⢰⣆⡋⣼⣶⠦⢶⡶⢧⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠺⣾⢅⡀⠀⢺⡦⢀⢐⣏⡷⡎⢻⢼⣞
⣿⣿⠛⡿⠊⡓⠚⣧⡇⣼⣡⠚⣶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣣⣇⡃⢺⡿⠬⠁⠘⣻⡧⠑⢈⣿⠞⡓⠛⡿⣴
⣿⣥⣷⠓⠋⢸⡏⠙⣿⣿⣧⣾⣿⣦⣿⣷⣷⣴⣶⣦⢤⣤⠒⣧⠯⢝⣄⡀⣀⢠⣌⡠⢥⠘⢄⢴⣯⣿⣦⣿⣈
⢸⣿⣟⠉⡱⢋⠙⣟⢻⡿⡿⣿⢻⢿⣿⣿⠿⠿⢶⣿⣿⡯⠔⣣⠘⢦⠷⣿⢇⡨⡚⣧⢉⠄⣍⢸⣿⣿⣿⠿⢿
⠘⠹⢏⡨⠦⠈⠓⠛⣍⡁⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠩⠃⠁⠭⠀⠉⠠⠓⠐⠀⠀⠠⡇⠘⠀⠓⢊⣃
⠀⠐⠒⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠂⠂⠠⠤⠤⠤⠤⠀⠄⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⣶⣤⠠⠄
⠀⠀⠀⠀⠀⣀⣰⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠈⠡⠔⠂⠀⡴⠆⢞⠿⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠤⠤⠤⠤⠷⠾⠳⠲⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠠⠴⠶⠛⠡⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿