Watercraft Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠁⢸⡇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⣿⠀⠸⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡀⠀⠀⠀⠀⣿⠀⠀⠠⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢴⡿⢴⠀⠀⠀⣿⡇⠀⠀⠀⠀⠣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠀⠀⠚⠁⠀⠀⠐⠀⠀⠑⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣴⣠⣶⢆⣠⠅⢀⣴⡀⠀⢂⣀⣠⣽⣾⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⢿⣯⣿⢾⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠠⠀⠉⠀⢀⣥⣶⣿⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⣶⡆⣤⠶⠚⢉⣩⣻⢟⣿⣿⡟⢿⣿⣿⠿⠛⠋⠉⠉⣹⢿⣿⣫⣔⡲⢶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⣷⣒⣩⣤⣾⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⢠⠀⡴⠻⣺⣿⣿⣟⢏
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣶⣶⣿⣿⡿⠿⠟⠛⠉⠀⢀⠀⠀⠀⣀⣥⠄⠀⠈⣀⢰⣟⠀⠀⠈⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠤⣄⣶⣤⣀⣀⠀⠀⠀⠀⢀⣠⣦⢾⣿⣿⣿⣿⣟⡛⠁⠀⣰⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⡿⠉⣠⣬⣿⣿⠟⡡⠋⢡⠾⠿⠛⠻⠷⠢⠀⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣝⠿⡷⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣶⣶⠒⠒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣠⣿⣶⣷⣾⣿⣶⣿⣶⣿⣾⣷⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢺⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⢻⠀⣀⡼⠀⠀⠀⠠⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⡀⣿⣤⣬⣯⣦⣥⣧⣄⣢⣽⣶⡶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣟⣠⣼⣿⣿⡏⣠⣤⣶⠿⣟⣿⣿⣿⣿⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣟⣿⠟⠉⠀⢰⣿⣿⣿⣿⡏⢻⣤⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤⡀⠀⠀⢉⢿⣿⣿⣿⣹⠇⢀⣤⡾⠟⠉⣼⣿⣿⢀⣼⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⡿⠀⢀⣰⡿⢛⠁⠈⣩⡶⠋⠁⣀⣸⣿⣿⣿⠇⣾⣿⡯⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣼⣿⣷⡾⢛⡅⢸⣿⣿⠛⠁⣴⣾⣿⣿⣿⠟⠛⣿⣷⣿⣿⠟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠍⣽⣿⣎⣾⣿⣿⣾⣿⣭⣅⣠⣼⡟⠛⠁⣀⣅⣴⣿⣿⣿⣿⣿⣦⣀⠀⠖
⠀⠀⠀⠈⢿⣦⣵⣦⣀⣤⣀⣶⣶⣦⣌⠙⠿⣿⣿⣤⣿⠏⠙⠻⢿⠿⠟⠛⠉⠀⠀⠀⠈⠉⠃⠀⠙⠛⠋⠻⣿⣯⣅⣀
⠀⠀⠀⠀⠀⠉⠋⠉⠿⣿⠿⣿⢿⣿⣿⣿⡿⠿⢿⣿⡿⠟⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣋⣉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠱⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⠁⢹⡿⠛⡟⢻⣛⡟⣿⡿⡻⣿⢻⡟⣿⣿⢛⢿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠷⡶⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡇
⠀⠄⢤⠶⢤⡀⢠⣷⡤⢴⡆⣸⣇⣠⣤⡄⣐⣀⣠⣀⣠⣀⡀⡏⡀⠀⠀⣠⣰⡞⢁⠀⠀⠀⡄⠀⢀⠀⢸⣖⠀⠀⠀⠀⠀⠀⠀⢀⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠠⠹⣀⣼⣤⣬⣿⣦⣌⣤⣹⣯⣬⣭⠉⠿⣿⠙⣩⣹⣿⡗⢭⣶⣶⣖⣾⠻⣿⣿⣤⡼⠯⠷⣶⢸⠂⠛⠛⠠⠀⠤⠀⠀⡀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⣭⡇⠀⠀⠉⢙⠋⠉⠋⠁⠿⡟⠛⠛⣿⠒⠾⢿⣿⠟⠛⠛⢛⣛⣿⣿⣿⣿⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠂⠠⠴⠄⠠⠄⠀⠈⡈⠈⠀⠀⠀⠀⠀⠀⣄⡀⡤⠌⠙⡟⠛⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠇⠀⡇
⠀⠀⣼⢻⠂⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡤⠀⠐⠇
⠀⠤⠾⢿⣯⣭⡁⠈⠁⠀⠀⠀⠂⠀⠀⠀⣶⡶⣄⢰⣶⠦⠀⠀⠀⢠⣤⣧⣀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⣻⡎⠀⣀⣀⣀⣀⣀⣀⣤⡤⣤⣤⣶⣲
⠀⠀⠀⠀⢼⣿⡟⡉⣠⣀⣄⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣦⣀⡀⠀⠀⡈⠉⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠿⠿⢭⣭⣶⣶⠿⠿⠿⠿⠿⠛⡟
⠀⠀⠀⠀⠈⠛⠻⣤⣾⠿⢿⡾⠚⠛⠉⠉⠀⢴⠟⠛⠛⠉⠉⠁⠈⠉⠃⡤⠀⠀⠀⢀⣠⣿⣿⠿⠟⠓⠀⠀⠀⣀⡀⠀⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣶⣶⣶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣴⣾⠯⠉⠛⠛⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⠫⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⡟⠛⠿⠟⠋⠀⠀⠀⠀⠀⢻⡏⠉⠁⠀⠈⠉⠀⠐⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠛⠩⢤⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠤⠤⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠆⡀⠁⠀⠀⠀⢄⠀⠰⣄⠀⠀⠀⢀⣀⡀⠔⠊⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠃⠀⠁⠀⠀⠀⠐⢄⡑⠂⢈⣂⣠⠍⠊⠁⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠉⠉⠂⢐⡛⣀⠐⢄⠀⢀⡠⠴⣶⡾⠋⠉⠛⠉⠀⠀⠀⣀⠠⠔⢊⠕⠁
⠀⠀⠀⠀⠀⠸⣅⡀⠀⡠⠤⠂⠁⢂⣈⣤⢔⠊⠡⠖⠂⠉⠁⠀⠀⠀⣀⡠⢴⡲⠭⣒⠤⠚⠁
⠀⠀⠀⠴⠊⠁⠀⣀⠄⠁⣠⠖⠉⠉⠑⠒⠉⠀⠀⢀⣀⠠⠄⢒⠈⠍⣂⠬⠐⠊⠁
⠀⠀⠀⠀⠀⠠⠚⠁⢠⠞⠁⠀⠀⠀⠂⠀⠤⠒⢈⠁⠀⣠⠔⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠄⠀⢀⡠⠄⠀⡠⠊⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠊⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣠⣦⠀⠀⠀⢸⠃⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⠉⠉⢹⡆⠀⠀⢀⣠⣤⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣤⣿⣿⣽⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠉⠁⢀⣀⣀⡀⠄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⣉⣭⣝⣻⡷⠀⠀⢀⠀⠀⠀⠒⠶⠤⠐⠒⠂
⠀⠀⠈⣀⣤⣤⣴⣶⣿⡿⠿⠿⢷⣶⣿⣷⣾⣿⡟⣻⣿⣿⢻⣿⣭⣭⣵⣶⣶⣶⣿⣿⣿⣿⣿⣿⣽⣦⠄⠀⠐⠛⠛⠓⠒
⢀⣀⣴⣶⣦⣤⣤⣍⣿⣷⣿⣿⡿⢿⣽⣿⣿⣛⣚⣛⣿⣧⣴⣶⣿⢿⠿⠿⢿⣿⣿⣟⣻⣿⣿⣿⣷⡶⠶⠶⠆⠈⠉⠉
⠀⠁⢭⣉⣉⣋⣉⠉⠛⠳⢤⣄⣀⣭⡅⠉⠳⣿⣿⣉⡀⣾⣿⣿⡿⠉⠛⠉⠀⠀⠐⠺⣶⣚⣛⣿⣿⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠚⠓⠶⠦⣤⣤⡤⠀⠀⠈⠉⠁⠶⠶⢿⣿⠇⠀⠈⠉⠉⠉⠉⣉⣀⣤⠴⠆⠀⠀⠀⠋⠉⢉⠉⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠘⠛⠃⠀⠀⠉⠉⠉⠒⠒⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢼⡇⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⢸⡇⣷⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣿⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠉⠉⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣥⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠛⠋⠉⠉⠉⠉⢹⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡿⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣀⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡇⣿⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡾⠁⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⣿⠀⠀⢀⣴⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⢇⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣿⠀⣰⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣷⠞⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇⣿⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⣀⣬⣿⠦⠴⠶⠤⠤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⣤⣤⣀⣀⣀⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠶⠶⠶⠶⠿⠓⠒⠛⠛⢳⡆
⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠻⠶⠴⠶⠶⠒⠒⠛⠓⠒⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⣠⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⣿⣿⡆⠀⠀⣟⠓⠂⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⣼⣿⣿⣿⡿⣥⡀⠀⠈⠫⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⣿⡿
⠀⠀⣼⣿⣿⣿⣿⣇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⢀⣿⣿⣿⣿⡟⣽⣿⣿⣇⠀⠀⠀⠀⢠⡴⠟⢷⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⡄
⢰⣀⣀⣤⣼⣤⣼⣭⣭⣿⣿⣦⡀⠀⠀⠀⢀⣤⣾⣿⡟⢁⠀⠀⠀⠀⠀⠘⢷⣶⣶⣶⣶⣷⣿⣷⣤⣴⣶
⠈⢙⣋⣉⣿⣉⣛⣉⣝⣋⣙⠉⠀⠀⠀⣓⣫⣝⣛⣍⣛⣫⣑⣊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠉⠉⠀⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤
⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣀⠀⠀⠛⠛⠋⢩⠟
⢠⣄⡀⣿⡟⠁⠀⠀⣽⠀⠀⠀⠀⣀⣤⡄⠀⠀⣸⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣄
⠀⠙⠛⢿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠀⠀⠀⡟⣿⣾⠀⠀⠲⣶⣶⣤⣴⣶⣶⣴⣿⣿⣶⣶⣶⣶⣶⡶
⠀⠀⠰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⡁⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣇⣀
⠀⠀⣾⣿⣷⡀⠀⠀⣤⣭⣥⠀⠀⠀⠀⠀⠬⠴⠮⠭⠴⠮⠤⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⠙⠛⠟⠀⠀⠀⢻⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡦
⠀⠀⠀⠀⢘⡃⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣐⣚⣍⣓⣚⣉⣚⣀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⣠⡆
⠀⠀⢀⣴⣿⣿⣦⡀⣿⣿⣿⠀⠀⠀⠀⠀⠈⢁⣀⣉⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣾⣿⡤⠴
⠀⠀⠀⣿⡏⢹⣿⠀⣿⣿⣿⠀⠀⠀⠀⣀⣀⣰⣿⣭⣿⣿⣿⣿⡆⠀⠀⢲⣶⣶⣶⣶⣯⣥⣤⣤⣭⣥⣤
⣀⣤⣤⣿⣷⣾⣿⣀⣿⣿⣿⣧⣄⠀⠀⠈⠿⠿⣿⣿⣽⣤⣧⣤⡇⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠛⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⢂⢀⣉⣀⣀⠀⠀⢐⡤⢀⣀⣀⣉⣑⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣥⠞⣻⣿⣿⣿⠟⢀⣾⣿⣿⣿⣯⠤⠤⠤⠈⠀⠀⣈⣒⡒⠀⠒⢒⡒⠂⠫⠉⢩⠉⠉⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠊⢐⣛⣛⣛⣉⠀⠉⠉⢡⠀⣠⣄⣶⣲⣆⣴⣶⡧⠤⠤⠤⠤⠤⠤⠤⠄⣙⣛⣓⣒⣦⠷⠖⠂
⠀⠀⣀⣄⠀⣀⡤⣖⠯⠤⠤⠛⠓⠒⠒⠛⠛⠉⠉⠉⠉⠉⢉⣀⣒⣒⣒⣒⡂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠋⠁⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠈⠉⠉⠙⠻⣦⣶⣤⣤⣠⣤⣤⣤⣴⠷⢶⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠤⠀⠠⠤⠀⠀⠤⣄⣀⠠⣄⣀⡀⣀⣀⣤⣄⣀⣠⡀⢀⣀⣄⣀⣉⡷⠘⠁⠀⠀⠀⠈⠋⠉⠉⠻⣦⠤⣄

⠀⠀⠀⠀⠀⠀⠄⠀⠂⡀⠂
⠀⠀⠀⠀⠀⠐⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⢀⠀⣤⣀⠴⡀
⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠃⡀⠀⠈
⠊⠋⠄⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠑⠠⠄
⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢖⣼⠷⣄⠀⠖⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠂⠰⠶⠀⠀⠋⠁⠙⠊⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡤⠖⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠄⠉⠀⢠⡿⠃
⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠟⠁⠀⢰⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣬⣿⣿⣷⣶⣤⣶⣿⡟⠀⠀⠀⠀⠀⢈⣰⣶⣀
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⡠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠆⢶⣤⣦⠘⠿⠿⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠠⠀⠀⠀⠀⡼⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠃⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢁⣼⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣴⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣵⣿⣶⣴⣥⣤⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡿⢘⣫⠁⠀⠀⡀
⠀⠀⠀⠀⠤⠤⠄⠀⠐⠂⠀⠈⢪⣽⠿⡿⣿⣿⣿⣿⣷⣯⣤⣤⣁⡀⠀⠀⢠⣾⣿⣿⡏⠀⣀⠤⠠⠖⠓⠈⠉⠀⢸⣿⣿⠇⢸⡿⠁⠀⠈⠁⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠒⠀⠩⠭⠀⠠⢶⣯⣓⣒⠬⠭⢟⣻⣧⣿⢿⣿⣿⣿⣿⣿⣿⣏⣉⡀⠤⣤⠠⢖⠲⣟⠙⢋⣀⡄⣹⡏⠀⠈⡄⠀⡀⠀⠐⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠠⠥⠤⠥⠤⠤⣤⣤⣝⣿⣿⣿⣿⣳⣲⡦⠤⠤⣤⣤⣤⣤⣠⣀⣀⣀⠀⠠⠤⠑⣚⣢⣊⣽⣽⣥⣴⡿⢆⡁⠀⢣⠀⣀⠛⠛⢃⣀⣀⣀
⠀⠐⠒⠀⢒⣓⣒⣒⣒⣀⣀⣀⣀⠐⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⡄⠂⠀⣾⠤⠤⠄⠀⠀⣀
⠀⠀⠀⠀⣉⣉⣭⢭⣁⠈⠉⢉⣿⣿⣧⣤⣤⣀⡀⢀⣈⡉⣙⠛⣋⠉⠛⠛⠿⠛⠛⠿⡿⠿⠟⣿⡿⠛⠉⡀⢀⣀⣀⡤⣆⡉⠀⠙⠋⠛⠒⠂⠀⠀⠁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠂⠉⠁⠀⠁⠈⠁⠀⠸⣄⣤⣭⣓⠢⠘⡛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢄⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣴⠟⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢉⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡵⢟⣵⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢸⠀⠀⠀⠀⠀⣀⡴⣻⡷⣫⣴⣿⣿⠟⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⢸⡆⠀⢀⣤⠞⣫⠞⣫⡾⣋⡵⢋⡴⣊⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣧⠾⢋⡵⢊⡵⢛⡥⢞⡡⢚⡡⢚⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢹⡞⣩⠖⣫⢔⡭⢒⣡⠖⣩⡶⢋⡵⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢻⡵⡫⢞⣡⠞⣋⣴⠞⣩⢞⣡⠞⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢞⣻⠀⠀⠀⠀⠀⠀⠀⠀⢋⠔⢋⣠⠾⢋⠔⣫⠴⣫⡵⣛⣵
⣛⡻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣟⠿⣒⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢋⡥⢚⡵⢊⡵⢞⣥⠾⠛⠛
⢯⣝⡓⠦⣝⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⢟⣋⡥⠶⣛⣩⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢞⣫⢞⣯⢞⣋⣥⣶⠾⠟
⠶⢬⣙⡳⢬⡛⠦⣍⠻⢶⣞⡛⠛⠻⠭⠝⢛⣋⡩⠤⠒⣊⣩⣤⣶⡿⠿⢛⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠯⠞⠛⢉⣡⣴⣶⣿
⢝⡲⢬⣙⠲⣭⡳⢮⡙⢦⣉⠻⣶⣒⣉⣩⣭⡤⢶⣾⠿⠟⢛⣩⠥⢒⣋⡥⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡴⠶⠛⢋⣩⣴⣿⢟
⢶⣬⣑⠮⠙⠦⣝⠲⣝⠳⣬⡳⣌⠛⢧⡀⠐⠉⠥⢒⣚⣩⠥⠖⠋⢉⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠟⠛⣹⡿⣫⠞
⠒⠌⡙⠷⣦⡀⠀⢅⡈⡛⢿⣯⡲⣕⠈⠻⣄⠚⠋⠉⣁⡠⠴⠖⢛⣩⣵⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣤⣴⣿⢟⡴⢋⡴
⣛⠳⢦⣄⡈⠙⠳⣦⣌⠲⣄⢝⢿⣾⣟⢮⡈⠣⠈⠭⢀⣒⣤⡵⠞⠫⢥⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⢟⡵⢋⡴⠋⣠
⣿⣽⣶⣮⣝⡳⢦⣄⠉⠻⣮⡑⢿⣫⣻⣧⣽⠶⠞⠛⢯⣭⣄⡈⠙⠳⠦⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⢋⡴⢟⣵⠟⢁
⠶⠮⠭⣽⣭⣭⣦⠾⠷⠶⠛⠛⢋⠛⠷⠦⠬⣛⣛⠷⢶⣤⣉⠙⠻⢦⣄⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣸⡴⢋⣴⠟⣡⡴⢟
⠿⣷⢶⣬⡑⠲⠦⣤⣄⣀⣈⠉⠙⠛⠛⠛⠓⠶⠮⣭⣛⡚⠿⠿⣶⣦⣬⣙⣻⣤⣤⡐⢒⣒⠂⠀⠰⢯⣴⢟⣁⣴⣚⣉⣼⠀⡇⢚⣩⠞⣋
⠷⣌⠙⠶⢭⣟⡒⠶⠦⣭⣭⣙⣛⣛⠛⠛⠛⠛⠒⠲⠿⠿⠿⠷⣶⣿⣿⣿⣿⡏⣿⠉⠁⠀⠀⠒⠐⢒⡒⠀⠀⠠⠶⠾⠯⠿⠿⡯⠖⠛⣉
⡳⢬⣝⡶⣤⣉⡙⠳⠶⢮⣬⣀⣛⡛⠿⠿⣿⣿⣗⣛⣛⣲⣶⣶⣿⣭⣽⡟⠀⣧⢸⡀⠀⠀⠀⠀⠀⠨⠄⠀⣐⣂⣤⣶⣶⠶⠿⠿⢛⣛⣉
⣍⣓⠮⣭⣑⣻⡿⢿⣷⣶⣦⣬⣭⣭⣭⣭⣥⡴⠶⠶⢿⣿⡿⣟⣫⣭⠿⢷⣾⣾⡔⣧⣶⣢⣄⣀⣤⣶⡶⠿⠛⠋⣉⣠⣤⣶⡾⠿⢛⣋⣉
⡚⢿⣿⣿⣿⣿⡿⠿⠟⢛⣻⣿⣿⣷⣶⣶⣶⠶⠿⢛⡩⠤⠴⠶⠖⣒⣚⣋⣭⠿⢳⣯⣿⣿⠿⠛⠉⣀⣤⣴⣾⣿⣿⡿⠿⠟⠛⠛⠉⣉⣉
⢿⣷⣶⣮⣭⣭⣭⣍⣛⣿⣿⠿⠟⣛⣋⣡⣤⣴⣒⣂⣀⣀⣉⣭⣭⣭⠵⣊⣶⡿⠟⠋⣉⣠⣴⣶⡿⠿⠛⣋⣉⣡⡤⠴⠶⠚⠛⠛⠉⠉⠉
⢽⠶⠿⣿⣿⣿⡿⣩⣭⠖⠘⣉⡡⠭⠤⠶⠖⠒⠒⠲⠶⠶⠶⣶⣿⠿⠛⢉⣠⣴⠶⠟⢛⣉⣥⠴⠖⠛⠋⢉⣁⣀⣀⣤⣤⣤⣶⠶⠶⠶⠒
⣉⣩⣭⡭⠭⢒⣫⡥⠖⢚⣩⠥⠖⢒⣛⣋⣉⣉⣉⣉⣩⠿⠋⣉⣤⡶⠟⢋⣩⡤⠖⣚⣩⣥⣴⠶⠶⠟⢛⣛⣉⣩⣭⣤⡴⠶⠶⠶⠶⠖⠒
⠶⣴⠶⠟⣛⡩⢤⣒⣭⣥⣶⣾⣿⣭⣭⣭⣭⠿⣛⣭⡴⠾⣛⣩⣴⣶⣿⡿⠶⠛⣛⣉⣭⣤⣶⠶⠾⠛⣛⣉⣭⡭⠴⠶⠒⠚⢛⣛⣋⣉⣉
⠿⠷⠾⠫⠴⠾⠿⠿⠿⠿⣶⣶⣦⣴⣾⡿⠖⣛⣭⠵⣾⡿⠿⠛⣋⣭⡤⠶⠞⣛⣛⣭⡭⠴⠖⢒⣛⣭⣭⠤⠶⠶⠚⢛⣛⣛⣉⣉⣉⣉⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠂⠒⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠰⡧⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢢⡄⠀⠈⠀⢀⡇⠀⣀⠠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢍⣭⡝⢛⣒⢦⣈⣀⡀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠛⠀⠄⢠⡄⢈⣑⣠⣴⠎
⠀⠀⠀⠀⠀⠠⠤⠶⣤⣤⣤⡀⣀⣀⣤⣀⠘⣿⣿⣾⣿⣶⣬⣭⣽⣏⣉⣾⣀⣐⣃⣀⣶⣤⣤⣽⡶⠶⠛⢻⡟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⢐⣶⣶⣶⣮⣿⣟⣻⣿⣿⣿⣿⠿⣿⣿⠫⣿⣿⣧⠖⢦⣭⣭⣭⠨⢭⣥⣶⣢⣤⣿⣟⠛⢻⣶⣶⠏⢀⣒⠂⠀⠀⠁
⠀⠀⠉⠀⠀⠚⠤⠀⠙⠛⢛⡛⠡⠐⠺⠿⠶⠦⣀⣬⠛⠛⠿⠿⣒⡾⠛⠉⠠⣾⣿⣥⢂⠀⢩⡍⠀⣉⠉⠒⠂⠉⠉⠀⠀⠘⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠲⠂⠘⠛⠛⠻⠤⠀⠀⠐⠒⠀⢲⡀⠉⠉⠩⠤⠄⠀⠀⠉⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣭⣭⣽⣤⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⣶⣶⣤⣀⣀⠀⠀⠀⡀⢢⣀⠀⡀⠀⠀⠐⣆⡀⠄⠀⠀⢀⠦⠀⡷
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢀⣀⡠⣞⣠⡀⠀⣻⣿⣿⣿⣿⣿⣷⣶⣟⣡⣬⣿⣿⣅⣀⡀⢀⣈⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢀⣀⠁⢪⢁⣴⣦⣴⣼⣿⣾⣿⣿⣷⣶⣿⣿⠿⠿⢿⣿⣿⣿⣿⠟⠛⠛⠋⠛⠻⠲⢶⣶⣄⣀⣄⠽⡿⢆⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⢀⠉⢻⡿⢿⣿⡿⠛⠉⣁⣉⣀⣿⣻⠛⠻⣿⡿⣿⣟⠋⠛⠐⢻⣷⡜⠀⡹⠀⢉⠙⢿⣿⢷⣆⣠⠀⠉⡙⠟⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢿⣛⣛⢈⣹⠛⠛⠛⡧⡙⢀⣾⣷⣾⣷⣦⣤⡴⠼⠿⢿⣖⠓⠟⢺⡚⠿⡯⣤⠬⢨⡆⠼⡯⣤⣁⣀⣀⣁⡀
⠀⠀⠀⡠⠤⠀⠀⡄⢠⣤⡤⠈⠀⠀⠀⢀⡤⡀⣄⠡⠫⠿⠽⠟⠛⠛⢛⡛⠋⠩⠄⣈⣿⣾⣄⣤⣈⠣⡄⡀⢠⣩⣽⣾⣿⣿⡿⠯⠄
⠀⠀⠠⠤⠤⠤⠤⠄⠤⠷⣈⡅⢾⠐⠶⠖⠶⠷⠷⠖⠒⠒⠒⠒⠚⠒⠚⠉⠛⠋⠉⠉⠙⠛⢿⡿⣦⣄⣭⢉⣐⣈⣀⣀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠻⣇⢻⡷⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢁⣠⡄⠀⠀⠀⠒⠷⠈⡛⠉⠐⠓⢚⠛⠛⠿⠥⠐⠶⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢂⣀⣹⣐⣏⣸⣧⡄⢈⠅⡀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠽⠛⠁⠈⠁⠸⢣⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣶⣶⣶⣿⣿⡿⢯⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠛⠲⠶⡤⣈⡙⠛⣿⣿⡿⣿⣿⣟⠋⠛⠙⠛⠛⠛⠻⠃⠀⠈⠉⢉⠉⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢂⠀⢆⣧⣤⣈⣓⠀⠸⣧⣀⣀⣀⣀⠀⣠⣤⡀⠀⠀⣰⣦⣤⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠒⠂⠈⡄⣷⡂⠀⠈⢹⡆⠀⢀⣀⣈⡉⠉⠀⠙⠛⠓⠂⠘⠓⠂⠀⠀⠋⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠘⡀⣄⠀⠀⠠⣙⡀⠀⠓⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠃⠀⠐⠂⠚⠁⠙⠀⠀⠀⠈⢷⡄⢰⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉



⣶⣶⣶⢶⠶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣶

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣄⡀⢈⣛⡛⠿⣿⡿⠛⠛⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣽⣿⢿⣿⣿⣏⢻⣷⣶⣿⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠘⠿⠲⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣠⣿⢷⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣧⢀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠿⠃⣸⡿⠿⠟⠹⠗
⢸⣿⣿⣿⣿⣾⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣬⣿⣿⣟⠿⡿⠛⢻⣿⣿⠀⠀⠁
⢸⣿⣿⣿⣯⣽⣀⡈⠏⠉⠀⣤⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⡦⠤⢴⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠤⢤⣿⠇⢀⠀⠀⠀⠄⣀⣀⣀⣴⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⡿⠽⡿⠿⠛⠛⠉⠉⠉⠉⢸⡿⠀⡈⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣾⣇⣶⡇⠠⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣉⣀⣀⣸⣃⣼⣿⣿⣿⣹⣽⣯⣽⣿⣿⣿⣿⣯⣽⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠉⠉⠉⢩⣉⣉⣭⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡀⡀⢠⣐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣧⡀⠀⠀⠲⣟⣻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣂⣿⣿⣽⣟⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⣿⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⢸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠀⣿⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⡇⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡇⢠⣿⣘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⡟⣿⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⣿⡷⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠏⠀⣿⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⠀⡿⣟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⣀⠰⡿⠿⠿⠿⡟⠽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⠅⠀⠈⠁⠀⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣾⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀⠸⠽⠜⡀⣀⠀⠀⠘⡆⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⡁⠀⠀⠀⠀⠿⢻⣻⣿⢐⢸
⠀⠀⣠⣀⣀⡀⠁⠀⣠⢀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣒⠭⢝⡀⠀⠀⠃⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠸⠁⠀⠀⠀⠀⠀⢘⣀⣸⣿⣿⢐⢼
⠭⠽⠿⠯⠿⠿⠾⠽⠿⠈⠏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡟⠉⠉⠉⠉⠙⠋⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠛⠛⠙⢻⣿⢻⣿⣿⢸⢸
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⢀⠾⣦⠸⣿⣿⣟⣿⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠹⠛⠁⠀⠀⠀⠀⠀⢠⣰⠀⢰⣶⠀⢰⡇⠀⠀⠀⠀⢸⣿⣿⣷⣿⣷⣿⣿⡯⣿⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠲⠞⠫⠍⠥⠶⣺⠍⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⡘⠛⠂⣸⣿⡷⠤⠤⠤⢤⣿⣿⣿⣿⣿⣿⣿⡇⣿⠨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⢉⣉⣥⣶⣿⡼⣥⠭⠙⠟⢸⡇⢴⣿⣼⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡇⣿⡄⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡈⠉⠀⠀⠀⣀⣠⣴⡶⠟⣻⠍⢽⠛⠈⣉⣥⣾⣿⣟⣫⣴⡿⠿⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⣶⠾⠋⣉⡠⢾⠒⠉⠐⣂⣤⣶⠿⡛⣉⠝⣛⡟⠙⠉⠀⠁⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⣿⣷⣿
⠀⠀⠀⠀⣀⣤⡴⠞⠛⣿⣿⣿⠀⣷⣾⠋⠉⠳⠬⠤⠶⠿⠏⡭⠥⢿⣿⣴⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣻⣿
⣤⠶⠟⠛⠉⢀⣧⡤⠔⢿⣿⡏⣰⡿⢿⣿⣷⣷⣶⣶⣶⣞⣢⠖⢒⣿⣿⣿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠶⢾⡷⣿⢿⣿
⠀⣀⡤⠔⠚⠉⣇⠀⣠⣿⣿⡗⠞⢛⣩⡙⠛⠓⣒⣧⣬⣭⣥⣿⡿⢿⠿⠁⠀⠀⢀⣀⣐⡒⠐⠒⠒⠂⠀⠒⠒⢒⡒⠚⠛⠛⠛⠚⣗⣿⣿⣿⣿⣿⣿
⠭⠤⡤⣤⣤⣘⣿⣟⣩⠼⣿⢀⣤⠾⠛⣁⣴⣾⣿⣿⣿⣿⠟⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⣭⣭⣭⣭⣭⣭⣭⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⣠⡄⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣾⣿⣶⣿⣿⣿⣿⣿⣆⣴⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠿⢟⣫⣶⣿⣿⣿⢭⣽⣛⡝⠂⠩⣭⣛⡛⠻⠿⠿⠿⢟⣛⣫⣵⣿⠟⣿⡿⢟⡪⣔⣮⠳⢍⡙⠛⢫⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣑⣃⣀⣠⠄⣐⣬⣽⢟⣋⡭⠷⠦⢶⣢⣤⡉⢔⣲⣭⣿⣿⣷⣮⣭⣿⣿⣿⣯⣙⣶⠾⠫⢾⣿⣿⣭⣖⣿⡶⠿⢛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠙⠋⣉⣉⣭⣭⣅⣛⠻⢭⣛⣫⡴⢮⣽⠷⠾⠿⠿⠶⡾⢿⣿⠿⣭⣷⣶⣮⠭⠽⠿⠛⠛⣒⣙⣫⡵⠾⠛⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⠻⠷⠶⢖⡫⠽⣛⣴⣬⠭⣿⣛⠤⣶⠾⢝⣟⣭⣝⣿⣿⣾⣷⣽⣍⣋⣙⣀⣈⣊⣛⠽⣵⣒⠴⢞⣋⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣭⣭⣭⡭⢽⣓⠶⢿⣻⣶⣬⡵⢞⣛⣯⡥⢶⣿⣥⠯⠽⣟⣿⣶⠭⢭⣭⡉⣉⣩⣭⠭⢴⣞⡯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢭⣿⣶⣢⢌⣉⣑⡒⠒⠂⢘⣫⢭⠒⣉⣽⣖⣛⣩⣭⣽⣭⣭⣽⣛⠤⠮⣌⡕⠀⠚⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣭⣛⡻⠿⣙⣻⣤⣤⠈⠶⣯⣝⡻⣛⠭⣑⣻⣧⣶⣾⣿⡿⣛⡿⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⢿⣤⣞⠿⠯⠉⣭⣯⣿⡶⠾⠧⣬⣽⣿⢥⣼⣟⡻⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠟⠛⢭⡴⣬⠴⠾⠿⠟⠟⠋⠛⠈⠁




⣶⣶⣶⣶⣶⡾⢶⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣾⢿⡾⠶⡶⠶⢾⣶⡾⠾⠾⠶⠶⢾

⠀⠀⠀⠀⠀⠀⡀⠐⠢⢄⡀
⠀⠀⠀⠀⡀⠀⠁⠁⠢⠄⡈⠑⠢⠄⡀
⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠈⠐⠠⢄⠉⠐⠠⡀
⠀⠀⠀⠀⢆⠀⢀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠒⠠⣀⠙⠂
⠀⠀⠀⠀⠨⡀⠸⠧⡌⠃⠒⠂⠀⠓⠀⠀⢘⠀⠀⣘⡖⠠⡀⠐⠠
⠀⠀⠀⠀⠀⢿⠀⢁⠀⠂⠀⠢⡈⠉⠢⣀⠀⠁⠠⢀⠀⠀⠀⠁⠢⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡝⠂⠄⢂⡀⠀⠈⠢⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⠙⠠⢀⠁⠢⣀⠉
⠀⠀⠀⠀⠀⠀⠈⢌⠓⣀⡥⢀⡰⠄⠘⠂⠀⠀⠉⠁⠀⠀⠀⠋⠀⠀⠀⣀⡄⠑⠢⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠙⠻⣥⣠⠀⠄⠐⠂⠀⠀⠀⠂⢁⠀⡀⠀⠀⠉⠢⣀⠀⠀⡉⠢⠄⠀⠰⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠛⠲⢦⣀⡀⠠⡀⠒⠀⠑⠄⠑⠀⡀⠀⠘⠁⠀⡇⠀⠀⠀⢠⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠙⠓⠾⢦⣀⣀⠀⠀⠀⠀⠀⠐⠀⢀⠁⠀⠀⠀⡀⢀⡇⢀⣤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣄⣀⠀⠀⠀⠀⣂⣴⠶⠿⠟⠛⠛⠓⠛⠋⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢈⡑⢄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠳⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣼⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣤⣤⣄⠸⢆⡀⢠⠀⡄⠀⢠⠀⠀⢀⠀⠸⢾⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠦⠴⠂⢻⣿⣿⣿⣿⣷⣮⣑⠈⠐⠗⠒⠻⠒⠛⠒⠂⢰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⢿⣿⣿⣿⡟⠟⠁⠀⠈⡇⣤⢨⡀⠀⢰⣿⣿⣿⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠁⠈⣃⣠⣴⡶⣖⣶⣯⣶⠞⠋⠀⠈⠛⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⢴⣺⣽⢋⣡⡴⣟⣻⡟⠋⠁⠀⠀⠀⠈⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠅⢺⡏⢿⠷⠾⣶⣦⣤⣿⠛⠒⠒⠒⠒⠒⠒⠒⣲⣶⣲⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣯⡽⠇⣚⣵⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣯⣵⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣶⣿⣿⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠐⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠤⢸⡞⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢂⡇⢳⡐⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡅⢬⢧⠀⠀⠀⠀⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢸⠀⠀⠀⠡⠀⢰⠀⠁⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⠘⠀⠀⠀⠀⠐⠈⡇⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠤⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠭⠉⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⠄⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣠⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠙⢧⡀⡉⢉⢉⣉⢉⡁⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠼⠘⡇⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠈⢳⡌⠿⢻⠃⣫⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠵⠃⠖⣽⠀⠀⠀⠀⠀⠀⠘⢻⢳⡀⠀⠀⠀⠀⠑⡄⠁⠊⢁⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠈⢎⢿⣄⣀⠀⠀⠀⠘⢦⠀⠈⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⢀⠾⡇⢸⣀⣀⣀⣀⣀⣀⣠⣸⣮⣙⣛⠿⠷⣦⣠⣀⣳⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣚⣒⣒⣛⣒⣓⣘⣋⣿⣿⣿⣿⣿⣙⣉⠙⠛⠻⢿⣭⣽⣿⣷⣶⠾⠛⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠿⠿⠿⠿⣯⣿⣿⣿⣿⣿⣿⣟⣛⣻⣶⠶⢶⣶⣶⣶⣾⡿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣿⠏⠉⠡⠤⠿⠿⠿⣿⣿⣿⣶⠤⡬⠉⠉⠉⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠒⠀⠀⠒⠚⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣣⣿⣧⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡇⠀⠀⠀⠀⠐⠀⢸⡇⠈⠀⠀⠀⠁⠀⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡗⡀⣀⠀⠠⢄⣀⡈⢠⣶⠀⢀⡀⠀⠀⠀⠀⠀⠁
⠀⣤⡀⠂⠀⠀⠀⣶⣿⡟⠳⠾⠛⠿⡾⠿⠿⠶⠦⢬⣤⣄⣘⠛⠛⠛⠛⠛⠀⢦⠀⠃⠀⢸⡟⢰⡄⠀⠀⠀⠀⠃⠓
⢸⣿⣿⠀⠀⠐⠀⢸⣿⣷⡞⠒⠂⠂⠰⠐⡀⠀⠰⠀⠘⣹⣿⡷⠀⠀⠐⠀⣀⣸⡇⣥⠤⣾⡇⢸⡇⠀⠀⠀⠀⠠⠤⠀⠀⠀⠄
⢈⣙⣙⣀⣄⡀⠀⠊⠛⠛⠃⠐⣀⣀⣈⣀⡠⢬⣀⣀⣈⣙⣛⣻⣁⠤⠤⠐⠤⠤⠤⢬⠄⣿⠇⢾⡇⠀⠴⠄⠀⠀⠆⠄⠊⠠
⠈⠉⠉⠁⠀⠈⠉⠉⣍⣉⣩⣧⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣹⠋⢉⠁⢚⡁⠀⠀⠈⢀⣿⠀⢺⡅⠀⠀⠀⠈⠉⠉⠉⠛⠟⠉
⠀⠀⠁⠈⠈⠉⠉⠀⠈⢛⢛⣉⣀⡀⣀⣀⡀⠥⠤⠤⣄⣄⢂⣻⣛⣛⣩⣉⣯⣭⣭⠙⣹⣿⠀⢻⡇⠉⠉⣉⢢⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠞⠛⠛⠓⠶⠷⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠋⠉⠴⣾⡏⢠⣼⠇⣠⣀⣀⣠⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣶⣿⣿⣿⠿⠷⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣄⠁⢼⡇⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢠⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣵⡟⠀⠀⠀⡌⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣴⡿⠀⠀⠀⢀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢰⣿⠇⠀⠀⠀⡘⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⣿⡟⠀⠀⠀⠀⡇⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣿⡇⠀⠀⠀⢸⠁⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠄⢰⣿⡇⠀⠀⠀⡘⠀⠀⡅⢸⡟⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠘⣿⣧⠀⠀⢀⠇⠀⢀⠀⢸⣇⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣿⣿⡀⠀⢸⠀⠀⠈⠙⢸⡿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠈⢾⣿⣧⠀⡇⠀⢀⠃⠀⢺⣿⡄⢿⣿
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⣼⣿⣿⣶⠇⠀⢸⠀⠀⣼⣿⡀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⠀⠈⠾⣿⣿⡄⠀⡆⠀⠀⠈⣿⡇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡓⠢⡉⠁⠀⠀⠙⣿⣿⣀⡅⠀⠀⠀⣿⣷⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⡀⠀⡀⠘⣿⣿⡀⠀⠀⣐⣿⣯⠀⠀⢈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠉⠲⣿⠆⡙⣿⣷⠀⠀⢽⣿⣿⠇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠙⠻⢟⣿⣧⠀⠂⢸⣿⣷⣆⠀⢀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⠀⢀⠰⣿⣿⣿⣿⡷⡄⠘⣿⣿⣿⣶⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⢀⣠⣾⣿⣿⡿⠟⠃⠀⠘⡿⠛⠛⠛⠛⣛⣛⡛⠛⠛⢣⣶⣶⠂⠠⠄⠄⠀⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣯⣥⣤⣴⣦⣤⣤⣧⡄⠀⠰⠾⠿⣿⣿⣦⣴⣿⣿⣷⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣉⡉⠛⠿⠻⢿⡻⠿⠭⣭⣿⣟⣾⣟⣸⣯⣿⣭⣭⣭⣽⣯⣽⠿⠿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠿⠷⣶⣿⣿⣷⣾⣷⣷⣶⣦⣼⣧⣤⣤⣈⣋⣤⣤⣬⣡⣤⣤⣥⣤⣈⣀⣆⣙⣿⣾⣷⡶⣶⣶⣆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⢛⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠋⠉⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠘⠀⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣤⣤⠐⣲⡿⠁⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⠀⢸⣷⣮⣿⡯⣹⠋⠀⢀⢠⣤⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣾⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠢⡤⡘⣿⣿⣷⣌⢻⣿⣿⢽⢿⣤⣌⠁⣈⡿⠆⠀⠀⠀⢨⠁⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣫⡿⣿⣿⣷⣿⣿⡅⠁⠉⠉⠉⠉⠉⠉⠁⠀⠈⢸⠳⢴⢃⡘⠈⠻⠋⠑⠉⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣿⠟⠉⠓⠒⣲⠖⠒⠒⠒⠒⠲⡖⠒⠚⠢⡀⠙⠷⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⢀⡴⠞⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⢄⣀⢝⢶⣫⣤⣾⣿⣶⣤⣤⢀⠀⢀⠀⢀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⡇⠄⣩⣵⣶⢻⣻⣷⣾⢣⣿⣿⣿⣷⣷⣷⣷⣾⡏⣽⣿⣶⡆⣴⠀⡦⢤⡄⢠⡶⠀⠀⠀⡀⠀⠘⢿
⣿⣿⣿⣿⣿⣿⣿⣛⠿⠟⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠾⠁⠺⠦⠻⣅⠠⠔⠈⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣿⣿⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡞⠀⣿⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⣠⣿⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⡀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠿⠿⠛⢿⠀⢀⠀⡀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠙⠛⠛⠛⠛⠛⠉⠉⢸⡇⠀⠀⠀⠀⢀⠀⠡⡀⠀⠈⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣁⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡤⠴⠒⢛⣿⠏
⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⡇⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⣰⣟⣥⠶⠒⠒⠶⢤⣀
⣀⣀⣀⣀⣠⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠷⠶⠒⠚⠛⠉⠉⠉⠀⢀⣀⡠⠴⠒⠂⠀⠀⠀⠀⢀⡼⠟⠛⠛⢷⡄⡠⠤⢄⠙⣦
⠙⢿⣿⣿⣿⡛⠛⠛⠉⢉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⣾⣡⡾⢻⠀⠤⠼⠃⣿
⠀⠀⠙⢿⣿⣿⣦⠠⠤⠼⢦⠤⠤⠤⠄⠒⠒⠒⠒⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⣀⣠⠤⠖⠂⠀⠀⠀⣀⡤⠾⠷⠖⠲⠶⠶⣤⣤⠤⢼⣦⣤⣤⠾⠗⢦
⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⣤⣤⠤⠤⠤⠴⠒⠒⠋⠉⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⣠⣤⣄⡀⠀⠀⠀⠙⣷⡄⣀⡉⢳⡄⠘⠾
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠊⠁⠀⠀⠀⢀⡾⠁⢀⣬⣷⠀⠀⠀⣰⡿⠱⣧⣠⠾⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⢤⣈⣉⠛⠓⠲⠶⠤⠤⠤⠤⠴⠶⠖⠒⣛⣩⠽⠛⠉⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠉⢁⣠⣴⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠀⠀⠁⢀⠀⠁⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣀⠀⡀⢠⠀⠶⡏⠚⢨⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠐⠋⠈⠂⠀⠂⠀⠀⠁⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡤⠔⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⢤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⠀⠐⠺⠀⢁⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⠖⠋⠁⠀⠀⠀⠀⠉⣄⠀⡉⠀⠔
⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠠⡀⠐⠒⠤⠂⠂⢑⡒⠚⠛⠛⠛⠛⠛⣻⣾⣥⡀⠀⢀⡄⠠⡼⠱⢨⠃⠀⠀⠆
⣠⣴⠯⠒⠀⠀⠀⠂⠠⠀⠁⠀⠀⠀⢀⣀⣤⣎⡀⠀⡀⢋⣁⣤⣾⣿⣿⣿⡿⠀⠀⠀⠀⠑⠂⠼⢀⡀
⠀⠀⠈⠒⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠠⣀
⠀⠀⠈⡒⠒⠛⠀⠀⠀⠒⠤⢄⣀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠈⠉⠀⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠒⠒⠐⠂⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢅⠀⠐⠀⠀⠀⡀⠀⠔⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠢⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⠀⠐⠁⠠⡀⠀⠄⠐⠠⢄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠄⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠑⣀⠌⠀⠀⠀⠈⢀⣤⣴⣶⣶⣿⡿⣷⣶⡶⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠴⠶⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⡿⠷⠿⠿⠿⠷⠤⠄⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⡐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⢀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡏⠙⣷⢸⣷⠀⡖⣿⠁⢠⣿⡇⠀⡦⠉⣿⠁⣾⡇⠀⣸⣼⣿⠉⢷⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⠀⣿⠈⣿⣴⠇⣿⠀⢸⣿⣿⢸⠇⠀⣿⠀⣿⡇⢀⡇⣿⣿⣠⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣿⢸⣿⢀⡟⠀⢻⣿⠀⣿⠖⢸⡏⢻⣿⠀⠀⣿⠀⢻⣧⡾⠁⢿⣿⠻⡄⢸⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠋⠉⠀⠀⠈⠉⠀⠙⠃⠈⠁⠈⠁⠀⠀⠉⠀⠈⠉⠁⠀⠈⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣀⣀⣀⠀⠔⢢⠀⠀⣀⠔⠂⠀⡃⠀⠠⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠈⠑⠑⠀⠁⠊⠁⠀⠊⠐⠈

⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⣼⡏⠀⣠⣄⣠⣿⣄
⠀⠼⠮⣿⣬⣷⣿⡿⣿⡈⠀⠀⠐⢀⠀⡀⠀⡀⠀⠀⠀⢀⣤
⠘⠀⠀⠀⠀⠀⢈⣱⣿⣿⡃⠠⡀⠙⡀⣀⠆⡠⢄⣀⠀⣾⣿⣿⣧⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠛⠀⠀⢐⠂⢸⣿⣿⣯⣽⣷⠛⢒⡠⣣⡿⠟⠛⠉⠛⠶⢄⡈⠙⠐⢿⣄⢀⠀⠀⠐⢶⣮⣷⣿⣴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠛⠿⠓⠛⠛⠷⠮⣿⣯⣤⡐⠒⠒⠒⠒⣚⣛⠗⡆⣀⢈⠹⠆⠐⠶⠾⡷⣽⣿⣿⣶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢤⣿⡇⠀⠁⠉⠈⢠⡤⠄⠄⠀⠤⠄⡽⠍⡘⠋
⠈⠁⠆⠀⣿⡇⠈⢿⠿⠻⣿⠿⠗⠿⡿⠶⠛⠟⣷⣻⣿⣿⠻⠃⠀⡄⡤⠠⠘⠁⡆⢠⠠⠄⣸⣧⡍⠉⠁⠁⠚⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡄⠀⠏⠈⣇⠸⠀⠖⠋⠛⠛⠃⠀⠀⠀⢀⡁⢀⡄⠀⠁⠀⠀⠀⢡⡄⠀⠀⠀⠐⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡁⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠓⠶⡶⢆⣼⡷⡲⠒⣠⣤⣬⡝⠀⠐⠒⠀⠀⠀⠀⠒⠀⠘⠉⡭⠑
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⣠⡭⢤⢠⡀⠮⢾⡉⠐⠀⠈⠍⢐⡍⠀⠿⠹⠀⠀⠋
⠀⠀⠀⡀⠀⠀⠀⡀⣀⢤⠤⢰⠆⠀⢠⠀⡀⠁⡤⢴⠆⢹⣭⣽⠀⠃⠃⠘⣿⠐⠀⠛⠐⠂⠒⠒⠛⠿⠆
⠀⠀⠀⠠⠄⠘⠋⣤⣤⣶⣦⠬⢤⣀⡘⠀⠀⣌⠁⠉⠁⠈⣍⠉⣁⠀⠀⣀⡛⠦⢀⡝
⠀⠀⠀⠀⠀⠈⠀⠚⠿⡏⠯⡄⠀⠘⠛⠛⠛⠛⠓⠀⠙⠚⠻⠒⠀⠙⠛⠋⠋⠀⢀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠝⠛⠛⠛⠛⠀⣹⡟⠀⠀⠀⠀⠀⠂
⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣂⣯⣭⣭⣭⣭⣭⣬⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠜⠀⠀⠀⡸⠉⡹⠁⡹⠁⢸⠁⢩⡏⠉⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣾⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢿⣶⣤⣠⣃⢠⣃⣀⣇⡀⣘⣀⢸⢀⡀⣣⠀⠀⠀⠀⣿⣿⣿⣿⣧⡈⠉⠻⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣶⣤⣴⣦⣤⣦⣤⣴⣤⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢀⣀⠀⠀⠀⠀⣀⣠⡀⠀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⢿⣼⣟⣻⣿⣿⣟⠃⣰⢚⣻⠿⢯⣤⣿⣿⣿⣾⣿⣷⠟⣿⣿⣿⣿⠿⣷⣶⣦⣤⡖⣶⣶⠤⠤⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢛⣋⣈⣉⣀⣚⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣼⡿⠿⠿⠿⠓⠫⢽⡵⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣻⣟⣉⡉⠋⠈⢩⠄⠉⠀⠠⠍⠁
⠀⠀⠚⠛⠛⠀⠀⠀⠀⠀⣀⣴⣿⣣⡤⠤⢾⣷⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣾⣿⣿⠿⠷⠾⠿⠷⠤⠦⠶⠛⠿⠓⠂
⠀⠀⠀⠀⠀⠐⠂⠂⠐⠚⠉⠿⠟⠛⠉⣡⣼⣥⢐⣆⢻⣇⣽⡫⢽⡿⠟⠿⠿⣧⣭⢉⣥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡠⣤⣤⠴⠀⠀⠀⠀⢈⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉
⠀⠀⠈⠱⠀⠉⠀⠁⠉⠓⠂⠂⠀⠀⠀⠛⠛⠛
⠀⠠⠖⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⣠⣴⣶⣤⣶⣶⣦⣤⣶⣶⣦⣶⣶⣿⣶⣶⡶⠖⠂
⠀⠠⠄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠉⠉⠉⠉⠉⠸⠿⠿⠟⠛⠛⠻⠛⠛⠙⠛⠛⠿⢿⡟⠿⠿⠷⠂
⠀⠀⢰⡓⣉⡀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⣺⣷
⠀⠀⠌⠉⠛⠹⢃⡠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⣄⢸⣿⠀⣴⣦⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⣀⣾⣇⣀⣄⣀⣠⡶⢤⣤⣀⣀⣀⣴⣄⣀
⠀⣠⣦⣾⣴⣿⣿⣿⣶⣾⣿⣿⣿⣶⣿⣿⣦⣼⣠⣠⣤⣄⣧⣤⣦⢀⡀⣄⣸⣄⠀⣿⡀⢹⣿⣿⣿⣿⣿⣿⠛⢻⢿⠿⡟⠋⣿⣿⣟⣿⣾⣻⡟⢿⣿⣿⣿⣿⡇
⠀⢘⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡤⣶⣿⣿⣿⣶⣿⣧⡆⣿⣿⣿⣿⣽⣿⣤⣾⣼⣀⣷⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠘⠛⠋⢻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠙⠛⠛⠛⠛⠛⠛⠿⠟⠉⠛⠿⡿⢿⡏
⠀⢠⣤⣄⣸⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣛⡿⢿⠛⣛⡛⡛⠉⠛⠉⠉⠁⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⣰⣟⣿⣿⣿⣿⣿⣿⣶⣦⣤⠬⣽⠀⠀⠀⠀⠰⣾⣷⣶⣶⣶⢶⣿⣿⣿⣿⣿⣯⡄⠀⠀⠰⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣤⣀⣀⡀
⠀⢺⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣿⡿⡿⠿⠀⠀⠀⠀⡍⠁⠉⠈⠀⠚⣻⣿⠿⠿⠿⠟⠁⠀⠀⠀⠿⣿⣿⡿⣿⣿⠿⠯⠯⠽⣿⢹⣿⣿⡃
⠀⠸⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣄⣀⣠⣤⣂⣨⣭⣤⣽⣿⠄⠉⠸⣭⣥⣽⣿⡿⠇
⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⢿⡿⠿⢿⣿⡿⢿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠁
⣶⣶⣶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠄⠀⣤⣉⡠⢨⢮⡄⠀⣵⢊⣀⢱⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⡎⡀⠌⠛⠈⠁⠔⠕⡒⣠⠇⠀⠀⠀⠀⠀⠈⠁⢒⡠⢷⡒⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣧⠔⢨⠣⠀⠀⠀⡠⠚⠚⠉⠙⠣⢄⠤⣔⣌⡋⢛⡢⣀⠑⠃⠀⠈⠑⠲⣇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣍⡠⢊⣀⠀⠀⠉⠀⢀⢎⠑⡄⠀⠀⠀⢀⡜⠒⢏⠢⣑⣒⣚⠢⠝⠛⠀⠀⠸⠜⠀⠑⠤⡆⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡠⠯⣄⠘⠐⠁⠀⠀⠀⠸⣀⡀⣰⢀⣀⡴⡋⣀⡀⠀⠁⠒⣒⠊⠙⠂⠀⠀⠀⠀⡤⠤⣀⣠⢜⣦⣸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠝⣧⠟⡾⠀⠀⠀⡔⠓⡄⠀⠀⠀⠈⣁⠧⠴⠶⠒⠘⠦⠤⢄⡓⠃⠀⡰⠒⢢⠀⠀⡄⠁⡠⠉⢷⡠⠵⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢳⢣⢪⠁⠤⠃⡸⠀⠀⠈⠊⠁⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣘⠀⠊⠀⢀⠾⣉⠀⠀⠀⠏⠀⠙⡉⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢘⣤⠎⠀⢑⠋⡶⠈⠀⡤⡄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣌⡐⡹⢳⠤⠤⡸⢀⣀⠰⡴⢄
⠀⠀⠀⠀⠀⠀⢀⠁⣘⠀⡰⠋⠈⢣⠀⠀⠁⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⠛⡟⠁⠁⠀⠀⡰⠚⠉⡝⣧⢀⠇
⠀⠀⠀⠀⠀⠀⡠⢛⠛⣬⠁⠀⠀⠀⢦⠤⢄⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⠐⢍⡹⠀⠦⣐⠄⢡⣀⡏⢆
⠀⠀⠀⠀⠀⠀⢣⣼⢆⣹⠀⢀⣀⣀⣰⣋⢔⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣒⠀⢠⠄⢄⠀⠐⡤⢇⣁⡘
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⡏⠤⠀⢠⠗⢒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠘⠄⠼⠀⠀⣀⠀⣙⢱
⠀⠀⠀⠀⠀⠀⡱⠖⢠⠀⡜⠦⠴⠃⠀⠣⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠔⠒⢄⠀⠘⠦⠃⠴⣎
⠀⠀⠀⠀⠀⠀⣇⢰⠀⠉⠀⢔⣢⢀⠔⠒⣤⡞⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡀⢠⡁⠀⠘⢈⣆⣀⡀⠀⡇⠈⠆
⠀⠀⠀⠀⠀⠀⢀⠟⡀⣎⢉⡆⣀⠜⠋⡱⣡⠔⠋⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠞⠂⣛⠀⠉⠉⠀⢸⠠⠰⣰⡤⠊
⠀⠀⠀⠀⠀⠀⠘⢄⣱⠀⠁⠀⡉⢄⣣⠚⢃⠀⠀⢈⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣜⠾⠬⠹⡀⠀⠀⢀⡞⢢⣳⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢡⠀⢀⢈⠉⠉⠙⡇⠈⠁⠚⢆⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⢀⡀⡀⠰⣹⠂⠱⠤⡔⠋⠈⠒⣯⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠣⡣⠄⠃⢀⠤⢶⠀⠀⢠⠊⠉⢣⠀⠈⠙⠒⠒⠄⢤⠠⡴⢲⠚⠉⠀⢀⢀⠘⠤⠜⢀⡠⠂⢙⠶⢇⠀⢀⠌⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢱⢆⡒⣡⣂⠉⠑⢤⣤⣐⣂⡀⠇⠰⣉⠆⠀⡤⡄⢣⡛⢌⡢⡀⠀⢇⣀⠇⡠⡰⢀⢔⡉⠠⠞⢸⡠⠺⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠑⣧⠒⠊⠝⠋⠀⡀⠀⠀⠀⠀⡀⢴⡒⠍⠁⢆⠑⡄⠴⠶⠤⠤⢀⠀⠑⣡⢋⠌⡔⠉⢉⣽⣁⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⠑⣦⡀⠀⢏⡨⢀⡀⣀⢜⠴⢉⠻⣇⢀⠔⠁⠀⠀⢂⠀⠄⢠⠃⡄⢄⠉⠉⣠⠖⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⡀⢑⡦⢠⣊⡠⠤⢊⠒⠢⡽⠤⠧⣀⠀⠀⢀⣈⡥⠔⠁⠀⠑⣋⠔⠊⢸⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠠⠽⠒⠣⠤⣀⢀⣀⣀⣀⣹⣭⣋⡽⠶⠄⢶⡈⠁⡙⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠉⠒⠀⠚⠙⠤⠤⠞⠑⠒⠒⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠶⢢⣴⠄⠀⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢸⡀⠀⢀⡀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⠢⣀⢸⡇⠀⢸⣿⡆⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠈⠆⠈⣻⡇⠀⢰⠟⠀⠀⠱⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⢸⠐⠀⠀⠀⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡈⠀⠀⠀⠀⠀⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠥⢧⠁⠀⠀⠀⠀⠀⢿⢏
⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠀⠀⣸⣯⡝⠀⠀⠀⠀⠀⠀⣸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢅⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣧⢰⠀⠀⠀⠀⢀⡄⠋⠁⠀⠁⠁
⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠐⠶⣿⠿⡚⣥⡤⠄⠀⠀⠀⢸⣿⡈⠀⠠⠄⢀⣤⡿⠁
⠀⠀⠀⠀⢠⡿⢧⠖⠡⠤⠤⠤⠦⣿⣿⣿⣿⠀⠐⠃⠀⠐⢚⡻⢛⣤⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⢻⠛⢻⣦⠀⢀⣠⣼⣿⣿⣿⠿⠛⠛⠿⠁⡸
⣤⣤⣄⡀⢀⠀⠀⢀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡵⠾⣂⣰⡆⣸⣿⠀⣤
⣿⣿⣿⠛⠁⠈⢑⡀⠿⠦⣬⣛⣿⣿⣿⣿⣿⣟⣻⣿⣿⣧⣀⣤⣾⣿⣿⣿⡅⣿⠋⢹⣿⣷⣦⣤⣤⣖⠅
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣄⡙⠻⠿⠿⢿⣟⣿⠿⠛⠉⠉⠙⠻⠍⠙⠛⠋⠁⠈⠀⠘⣿⣿⡇⠉⣿⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠄⠈⠋⠁

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠴⠶⡒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣿⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⢤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠘⠛⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣇⣦⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡷⣄⡘⣦⠻⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢉⢻⢿⣿⣿⣿⣡⡱⣾⡝⢦⡈⠛⠛⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠈⠫⠳⣄⠻⢿⣿⣝⣾⣿⣷⠙⠲⣴⣷⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⣛⠛⠛⠉⣠⣴⣾⣰⣤⡴⠶⣤⣀⠀⠉⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⡿⡋⣠⣦⡈⡛⠿⣷⣾⣴⣴⣴⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠉⠉⠉⠉⠉⠉⢙⡿⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣼⠃⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠉⢳⡄
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠾⢧⣤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣧⣠⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⣼⠁⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢷⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣸⠃
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣰⠏
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣄⣀⣸⣇⣀⣀⣀⣀⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⠛⠛⠿⠿⣶⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣭⣭⣭⣭⣭⣿⣿⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣛⣉⣥⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣡⣴⣾⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⣿⡿⠿⠛⣋⣉⣤⣤⣶⣶⣶⣶⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⠉⢉⣉⠉⢉⣉⡉⢉⣁⡀⢀⣈⡉⠉⠉⢉⣉⠉⠉⣉⣉⠉⢉⠉⠉⡉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣻⡏⣵⠿⠟⣳⣿⡛⣡⣿⡛⣱⣟⣽⠟⠀⣴⣟⣿⢫⣞⣹⢟⡴⢿⡟⢠⡿⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣲⣶⣶⣦⣤⣽⣛⣄⠀⢀⡄⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣀⣈⡉⢡⣤⣤⣤⠭⢿⣿⣤⡟⠀⡀⢺⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⣁⣤⣶⣶⣦⣤⣹⡇⣼⡇⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠟⠉⠸⡟⢙⡿⣦⠟⠓⠡⣿
⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠋⣶⢠⣾⠁⣼⠿⣿⣞⡻⣶⠾
⠀⢰⣾⣏⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣳⣿⣿⢋⣰⠋⠀⣿⣇⠛⠉
⠀⠀⠀⠴⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⢻⣿⣛⢿⠉⠓⠀⣿⠇
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠏⠀⠀⠀⠀⠛⠁
⣀⣀⣀⣀⣨⣬⣥⣤⣤⣤⣤⣤⣶⣶⣶⣦⡆⠰⣶⡶⢶⣶⠶⠾⠟⠋⠉⠉⠉⠉⠯⠿⠛⠏⠐⠂⠀⠈⠋⠉⠉⠉⠉⠉⠉⠁⡟⠀⠉⠉⠉⢠⡏⠀⠐⠀⠉
⠙⠷⠮⣭⣭⠟⠟⠻⠛⠿⠿⠟⠋⠉⠉⣉⣁⢀⣉⢉⣁⣀⡀⠀⠀⣀⠀⠀⠰⠀⠠⠞⠠⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠛⠻⠿⠿⠻⠿⠛⠋⠁⠀⠀⠐⠂⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣶⣶⣶⣶⡴⠒⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣲⣿⣏⡭⠉⠀⠀⠀⢸⠄
⠀⠀⠀⠀⢠⣼⣿⣷⣤⣀⣈⣁⣀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠋⠉⠛⣋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠙⢛⣭⠭⠿⠻⢟⣳⠦⣤⣤⣤⣤⣴⣶⣇⣀⣠⢤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠩⠭⣍⣫⣑⣛⣻⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠈⠈⠉⠙⠛⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⢀⣀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡉⣿⣄⣁⣀⣀⣨⠈
⠀⠀⢂⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠀⠈⠉⠉⠁
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠈⠈⠙⣧⠴⠶⡄⠀⠀⠸⠷⠞⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠀⠱⠀⠀⢠⣤⣤⣤⣿⣭⣭⣿⣭⣭⣭⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣭⣭⣭⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⣿⣿⣿⡉⣻⣿⣿⡏⢙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠿⣶⣶⣶⣾⣿⣭⣿⣛⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⠿⣿⣿⣶⣾⣯⣭⣭⣭⣿⣿⣿⡭⠽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣯⣽⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⠖⠛⣛⡉⠉⡉⠉⠉⠉⠉⠁⠩⠍⠉⠉⠉⠩⠙⠒⠒⠒⠒⠲⠤⣤⣄
⠀⢠⠶⠒⠾⠛⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡨⠟⠁
⠀⠀⢳⠒⠓⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⢉⠉⠉⠍⠉⠉⢀⡠⠟⠉
⠀⠀⠘⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠉
⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁




⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⣀⣭⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⢠⣤⣌⣉⣉⡀⢹⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⣅⣀⠀⡀⠘⣿⣿⣿⣿⣿⣦⣄⠂⡀⠀⢸⣀⣀⣅⣀⣀⣀⣀⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠤⠤⣽⣿⣿⣿⣿⣿⣿⠟⠀⠀⢠⠀⠀⠀⠀⠀⢠⠰⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠀⠀⠀⠀⠀⠛⠙⠛⠉⠛⠁⠀⠀⠀⣀⡸⢘⣃⣶⠄⠀⠼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠘⠀⠀⠀⢀⣆⠀⢀⡀⢤⠴⠚⠛⣋⣴⣷⣧⠿⠋⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⣀⣠⠤⢺⣿⢫⣥⢒⠄⠤⠖⠫⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢩⠥⠐⣂⣸⡿⠚⣛⠛⣻⣟⣥⡾⠟⠉⠉⠉⠉⠉⠉⠉⠙⣛⣛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣴⢖⣾⣯⣮⠓⣪⣵⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣈⣩⣭⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⡿⢟⣫⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠶⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢼⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⠃⠀⢀⢀⠈⠀⣰⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⣀⡄⠀⢸⠃⠤⠈⠧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢠⠖⠀⠀⠀⠀⢀⠁⠀⠀⢀⢀⣀⣤⣴⠊⠇⠐⢚⡒⠀⠀⠸⠁⢤⠐⡦⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠀⠐⢢⠴⠐⠀⠀⢀⡀⢀⣁⢀⠐⠺⠻⠙⠋⠃⠉⠐⡇⠀⢀⡁⠺⠸⠇⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠋⣁⠀⠀⠈⢩⠇⢤⠰⠒⣛⠋⠋⠉⢀⡀⢉⢀⢰⢺⢴⢼⣣⣟⡼⠀⠂⠁⠀⠀⠀⢀⡀⠀⡄⢨⡅⢰⡆⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⣠⣼⠇⢠⠋⠀⢀⡆⠋⠈⠉⡇⠈⠄⠀⠀⠀⠀⠀⠙⠈⠈⢹⣿⣭⣽⣿⣤⠰⡆⢘⣿⡀⠃⠈⠃⠀⠇⠀⠃⠘⠃⠉
⠀⣠⠖⠲⣶⠀⡒⢀⢐⡆⡇⠸⠄⢉⣁⢘⠐⠚⡏⠀⢸⡆⠀⠀⠀⡇⡈⠉⠀⠀⠀⠀⠖⠾⠾⠻⢼⣼⣿⢿⡗⣯⠀⠀⠀⢉⠀⡀⢠⠀⢀⣀⢠⡄⢠⡄⢠
⠍⡙⡧⡋⠙⡂⣀⣀⠸⠃⡇⢈⣆⣈⡀⣸⢀⡀⣇⣂⣸⣂⣀⢀⣀⡃⠁⣠⢀⣀⣀⣀⡤⣤⡠⠤⠾⠥⡬⠾⠁⣡⠸⢇⡇⠸⠄⠳⢰⠆⠈⠟⠸⠇⢸⠂⢹
⠶⠞⠱⡷⠶⡿⠷⢤⣾⣴⠇⡵⣽⣯⣄⡉⡬⣤⣧⣴⢾⣥⣶⣂⣠⠥⡄⠀⠀⣀⠀⠀⢀⡆⠠⠀⠀⠶⣴⣶⠦⢦⣠⣼⠀⠀⠰⣶⢶⠄⢰⣖⢠⡶⠶⡆⡠
⠀⠁⠀⠀⠛⠛⠋⣱⣉⠷⢷⣶⢟⠒⠛⣋⡗⣂⣚⣿⡈⣉⠙⣛⣱⣾⢻⡟⣗⣛⣓⡚⠇⠯⠿⠲⢮⣍⠋⣟⡋⠗⢯⠶⣄⡀⠀⠉⢈⣩⠉⢉⣤⡝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠲⣿⠓⠋⠉⠉⢙⠒⠀⠉⠈⠉⠉⠀⠀⠐⠒⠲⠦⠬⡯⡝⡃⠀⠂⠾⠂⠁⠀⠢⠒⠦⠉⣛⠌⢏⡈⡉⠈⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢭⡀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠙⠁⠀⠈⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡃⠀⠀⠀⠀⡄⠀⢲⡆⠈⢠⡉⠳⠷⢶⣤⣤⡀⡠⢏⡠⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢲⠀⠘⡆⠀⣹⠃⠀⠀⠘⠛⠛⠃⡀⠉⠀⢀⣣⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⢀⠇⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠓

⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠍⠀⢈⡻⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢠⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⣸⣿⠖⠋⠉⢛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⡏⠉⠙⠛⠛⠉⠙⠛⠛⠋⠉⢉⢍⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⡇⢀⣼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡎⠀⠀⠀⠀⢀⣀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠙⠚⠚⠛⠉⠉⠁⠀⢀⢢⠃⠀⠀⠀⠀⠀⠀⣈⡟⠉⠉⠉⠉⠉⠉⢉⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⣸⠀⣀⡀⢤⠄⠀⢀⢣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠧⠤⠤⠤⠴⢶⣦⠀⠒⠂⣊⠜⢀⡐⠒⠒⠒⠂⠉⠉⠁⠀⠉⢸⠋⠀⠀⣌⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠃⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢀⠀⠠⢄⠀⠈⡀⠀⡸⠀⠀⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠌⢫⠉⠉⠙⠛⠋⠀⠀⠀⠐⠒⠈⠂⠀⡨⢻⠊⠀⠉⠀⢠⣷⣿⣿⣟⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⠀⠀⠀⡸⠀⣀⡠⠔⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⣀⣀⠤⠒⠋⠉⠀⢀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⢶⡒⠛⠒⠦⡉⠓⢤⣀⡴⠃⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠒⠒⠉⠉⠁⠀⠀⣀⠠⠔⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠘⢏⠉⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠒⠈⠵⢦⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠈⢷⠀⠀⠊⠀⠀⠀⠀⢀⣄⠈⠁⠉⠀⢠⣞⡆⠀⠀⠀⣀⠑⠉⠂⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠣⣀⠀⠀⠀⠀⠀⠑⠘⢀⣀⡀⠤⠤⠀⠐⠒⠈⠉⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡒⠀⠀⠀⠒⠒⠒⠒⠂⠈⠉⠉⠉⣁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠤⠤⠤⠤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠠⠒⠶⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠀⠀⢀⣵⡼⣶⣿⠀⠀⠀⣤⣤⣀⣀⣀⣄⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠒⠲⠶⣾⢟⢭⣴⣶⣿⣏⠁⠒⢶⡟⠀⡫⠅⠉⠛⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢉⠃⣾⣿⠟⠙⠓⠈⠐⢄⢻⡄⠋⠀⠀⠘⠞⣦⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠂⢀⣒⣤⣄⡌⢠⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡷⡀⢰⠚⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⣉⡠⠤⠐⠛⡭⡙⢀⢎⠎⠀⣦⡀⠀⠀⠀⣼⡟⠛⢻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⣻⣿⠀⠀⠀⠀⢠⠁⣌⠃⠀⠀⠈⢿⣄⠀⠀⠙⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠁⡀⣿⡏⣀⠄⠀⠀⣼⡞⣤⡴⠈⡲⢟⣛⠿⠄⠐⢲⣾⢎
⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⢸⣿⠏⢁⣀⢠⣤⡿⣿⠩⠔⠁⠀⠉⠉⠀⢢⣄⠀⢧⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣼⣿⠟⠉⠀⠀⠹⢷⡝⣥⠀⠀⠀⢠⣦⠚⠀⠹⣄⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⢠⢊⠄⢀⠀⠀⡅⡟⠂⠀⠀⢹⠏⠀⠀⠀⢻⣷⠀⠀⠀⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣿⣧⣤⠁⣠⠎⠀⠀⠀⠀⣤⣼⠿⣦⣀⡀⠀⠈⠓⠤⣀⠀⠀⠉⠻⢶⣖⡈⠒⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢝⠻⣧⣾⣧⣄⣀⣀⢀⣀⠿⢏⡀⠈⠓⢮⣳⠶⢤⣀⠀⠈⠁⠒⠢⢤⣈⠻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⠀⠀⢘⣶⣦⣴⡞⠁⠀⠀⢀⡞⠀⠀⠀⣽⢧⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠐⣹⢿⡇⣿⣷⡄⣀⣴⣿⣡⣤⣴⠤⠉⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⡘⠻⢿⣿⢀⠏⠈⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠃⠇⠀⠀⠈⠁⠀⠀⢶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⢿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣠⡾⢧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡿⢄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⣷⣤⣤⣤⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠩⠁⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⣀⣡⢴⠶⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠒⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿⣷⢸⠀⠀⠈⠹⠷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⢠⠆⢀⣀⣀⣴⢨⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠘⠿⠷⢖⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣂⣈⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠄⠀⠀⠀⠀⠀⠀⠀⠠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢂⣄⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣵⠀⠀⡀⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣢⡢⠀⠈⢂⠀⠀⠀⠐⠀⠀⠀⢨⡍⠀⠊⠘⠀⡀⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠄⠒⠄⠀⢈⢠⠀⠒⠒⠿⠐⠂⣛⠃⠚⠇⢀⣺⡿⠁⠀⠗
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⠀⢀⠂⠀⡈⢸⠈⠉⠉⢉⡅⠀⢿⠁⠈⠩⡠⢸⡁⠠⡢⠂
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠠⠂⠀⡰⠁⠈⢂⣀⣀⣰⢢⣸⣇⣀⣀⣀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢰⠁⠀⠀⣇⠀⢀⢾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣆⠀⢀⠤⠒⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠈⠁⠸⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠀⠉⠐⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠢⡄⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠁⠁⠀⠀⠀⠈⠐⠠⡀⠀⡀⠀⠄⡁⠢⢀⡑⡄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⡀⣀⠠⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣷⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⠉⠛⠻⠿⠿⠛⠛⠩⠚



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠲⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢷⡦⠤⠤⠤⠀⠀⢀⡀⠀⠘⢆⠀⠈⢆⠰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠉⠓⠢⠄⣀⠀⠀⢨⠁⠒⠮⢧⡀⠀⠓⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠓⠒⠒⠛⠛⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠑⣄⡤⠞⠁
⠀⠀⠀⠀⠀⠠⠒⠉⠀⢀⡠⠴⠒⠛⠛⣉⣉⣙⣛⣓⠲⠮⢭⣟⣳⠶⢦⣤⣄⣀⣀⣀⣀⣠⣾⡷⠖⠛⣉⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠀⠀⠀⠉⠉⠙⠓⠲⠭⣍⣓⠲⠦⠭⢭⣭⣭⠤⠤⠶⠚⠋⣁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠒⠒⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠟⠛⢛⣛⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⡱⣫⣭⣵⡎⣶⣦⣿⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⢽⣿⠟⣈⡩⠵⠟⠚⢉⣩⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠙⢯⡙⠛⠙⠛⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷⣦⣄⡀⠈⠳⡄⠀⠀⠀⠀⠀⡏⠙⠒⠲⠖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡏⠈⠙⢿⣿⣿⣿⣿⣿⣦⡀⠹⡆⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣟⣭⣭⠙⠦⣄⠀⠀⠀⠀⣾⣿⣿⣯⡉⠙⠲⣄⡀⠻⣿⣿⣿⣿⣿⣿⣦⣿⠀⠀⠀⠀⠸⣆⡤⠒⠋⣉⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡭⣀⡤⢄⡠⠒⢮⣓⢤⣀⣰⣿⣿⣿⡉⠙⠓⣒⣂⣙⣻⡉⠧⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⣏⣳⣚⡛⠉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠯⠵⠒⠉⠀⠀⠀⠈⠓⠦⣽⣿⣿⣿⣿⣉⠉⠉⣉⣨⠷⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠿⢭⣉⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣪⣭⡒⢤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠳⣄⣈⣻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠤⣤⣄⡀⢉⣒⣒⣒⣉⣽⣿⣿⣿⣿⣎⠻⠿⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠤⠤⠴⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⢰⣿⣿⣿⣷⣤⣀⣀⣀⣀⣠⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣶⣶⣶⣌⠛⠛⣛⣥⣶⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠚⠛⠻⢿⣓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣚⣉⣉⣀⣽⣿⣷⣤⣤⣤⣬⣿⣶⣾⣿⣿⣷⣶⣒⣒⣒⣒⣶⠶⠶⢦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢒⣛⠛⠛⠛⣛⣛⣛⣛⣟⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⣉⡉⢙⣛⡶⠾⣟⣿⠾⡷
⠀⠀⠀⢀⣄⠲⣿⣿⣿⠉⠀⠛⠠⢽⡾⠾⣿⣿⣿⣿⡇⠨⠟⣿⣿⡧⣿⣿⣿⣯⣽⣿⣿⣿⡿⠿⠟⢛⣁⠩⠭⢍⡒⠶⢛⣩⡴⠋
⢀⠤⣴⢿⣄⠑⣙⡻⣿⣷⣤⣀⡀⠀⠁⠐⢺⡿⢿⣛⡷⣤⡶⠟⠛⠛⠛⢛⡉⠉⠭⠅⣂⠒⠢⠍⠀⠐⣀⣉⣤⣴⣶⣿⠟⠋
⡟⡦⠽⢎⠻⣷⣮⠡⠠⢍⡛⠿⣿⣿⡷⠶⠒⠉⢁⠀⠄⠩⢐⣂⣈⣬⣥⠴⠖⠚⣉⣁⣤⣤⣶⣶⡿⢿⣛⣭⡿⠞⠋
⢳⣜⢄⠘⠳⠬⡻⢿⣶⣤⣈⡑⠀⠀⣀⣈⣤⣤⣶⣶⠾⠿⢛⣛⣩⣥⣴⣶⠾⢿⣛⣻⣿⣭⣷⡶⠿⠛⠉⠁
⠈⣿⠲⣍⡲⢤⣟⡳⢯⣝⡛⠟⠹⠟⢛⣋⣩⣭⣴⡶⠾⣟⣛⣿⣭⣿⣶⣾⣿⠿⠿⠛⠛⠉⠁
⠀⣏⣾⣿⣝⠳⢦⣬⣉⣥⣤⠶⢺⣟⣻⣯⡽⠷⢶⣛⡻⠯⠽⠓⠛⠉⠉
⢰⣽⣿⣿⡿⠛⠷⣮⣍⣲⣚⡛⠭⠍⠓⠒⠈⠉⠁
⣶⢶⣿⢿⠶⢶⣶⠶⢾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠓⠲⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡒⠒⠒⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇⠘⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡇⠀⢳⠘⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡇⠀⠸⡇⠹⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⡇⠀⢣⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡇⠀⠀⣿⠀⠘⡇⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡇⠀⠀⣿⠀⠀⣧⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡇⠀⠀⡇⠀⠀⢸⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⡇⠀⢸⠇⠀⠀⢸⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⠀⠈⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣿⡇⣸⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⣷⣇⣀⣀⣀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣖⣾⢯⣽⣶⣶⡿⠽⡿⠷⠖⠒⠛⠛⠛⢲⣾⣿⣿⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠵⠒⠚⠉⣉⣀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⠿⠟⠛⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣿⣿⡾⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⢉⣉⣀⣀⣤⡤⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣤⡤⠤⠤⠤⠤⠴⠶⠶⠖⠒⠚⠛⠋⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⢶⡂⠀⠀⠀⠀⠈⠉⠉⢩⡟⠉⠉⠉⠀⠀⠀⠐⣰⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⠒⠉⠀⠀⠀⢀⡀⠀⣷⠀⠄⠀⠒⠒⠀⠀⣞⠠⠠⢠⠀⠒⠒⢀⡞⠁⡀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⡉⠄⠂⠡⠂⠀⠠⠊⠁⣠⠁⣠⠿⠤⠀⠒⢂⠀⢀⠈⡛⠀⢀⠎⡁⠁⡀⠙⠢⢄⠈⠂⠠⠠⠤⣈⡆
⠀⠀⠀⠀⠀⢀⣴⣥⣄⡁⠀⠀⢀⡀⠴⡴⠁⠀⡔⠁⢀⠊⠀⡔⠀⠠⠃⠠⠂⡘⠀⢀⠎⡐⠀⡐⠀⡰⠀⠀⡿⢛⠁⠀⠀⠉⢻
⠀⠀⠀⢀⠔⠉⢀⠂⠀⢹⠖⠈⠜⢀⠜⠀⢀⠌⠀⡠⠁⠀⡐⠀⡠⠁⡰⠁⡜⠀⢀⠌⡰⠁⡜⠀⠐⢁⡠⠴⠓⠷⣄⠀⠀⠀⠎
⠀⠀⠀⡌⠀⠀⣀⣠⣤⠁⢀⠊⢀⠊⠀⢠⠊⠀⡐⠁⢀⠌⠀⡔⠀⡘⠀⠜⠀⠀⠌⠐⠀⡀⠄⠲⡎⠁⠀⠀⠀⠀⠸⠀⡠⠊
⠀⠀⠀⡇⢠⠞⠁⠐⢄⠓⣥⣠⣂⡀⣠⣃⣀⣘⣀⣀⣈⣀⣀⣀⠀⠤⠼⠀⠐⠚⡍⠀⣴⠀⣄⠀⠘⡇⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠸⡏⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠠⡛⠁⢰⢣⡄⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⢂⢀⠼⠓⠂⠈
⠀⠀⠀⠀⠈⠢⢄⡀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠀⠀⠰⢂⠀⠀⠀⣀⠠⠴⠃⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠂⠀⠚⠂⠒⠒⠒⠒⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠀⠄⢐⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⠂⣀⠀⠬⠍⠐⠐⠀⠈⠉⠁⢀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⣠⣀⡠⠄⠀⢀⣐⠂⠬⠉⠀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⢀⠠⠄⢐⡂⠧⠀⠂⠁
⠀⠀⠀⠀⠀⠀⢰⠀⣒⠂⠀⠠⠌⠐⡢⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡐⠀⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠤⠤⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠢⠉⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⡀⠠⢐⠂⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠀⠉⣀⣠⠤⠖⢚⠟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠶⠚⠋⠁⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠐⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⠛⠛⠉⠉⠉⠉⠻⢿⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣙⣢⡀⠀⠀⠀⠀⠀⠈⠓⠀⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠑⣄⠀⠀⠀⠀⠀⠀⠀⢢⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠂⣀⠖⠂⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠀⠴⠞⠠⠔⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢼⣿⣿⣷⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠠⠼⠬⠤⠿⠶⠒⠀⠋⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⡀⢠⣤⡄
⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣷⣶⡄
⠀⣀⡾⡝⠿⣿⣿⣿⡿⠏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⡶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣀⣤⡤⣤⣤⣤⣤⣤⣤⠤⣤⣤⣶⣶⣶⣶⣶⣶⡶⠶⠶⠾⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣦⣤⣤⣤⣭⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⡟⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⢸⠇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⢰⡇⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡏⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⠃⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡿⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣿⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⢃⣿⠃⢰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⢻⣟⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠶⠾⠛⠛⠛⢛⣛⣛⣛⣛⣛⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣛⣉⣉⣥⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡟⠉⢻⡟⠿⠿⠿⠻⢿⠻⠿⠿⢿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⣿⣿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡄⠀⠀⡇⠀⠀⡄⠹⢿⣿⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⣿⣴⠀⣿⣇⠀⢰⣧⠀⣦⣧⠀⢀⠀⠀⠀⢀⣷⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⢀⣿⣿⣿⡴⣋⣠⡕⠺⢿⢀⣿⣿⣤⣿⡄⠀⠀⢸⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⢀⠜⣉⣙⣿⣿⣿⣿⣵⣶⣆⡬⣿⣿⣿⣿⡇⠀⢄⠹⣿⡅⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⡙⡟⢁⣚⡟⢫⣾⣿⣿⣿⣿⣿⣿⡇⠙⣿⡟⢻⣿⡤⠀⠀⢹⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⣿⢀⣿⡿⠿⠀⡾⣿⡷⢮⡽⠿⣿⡿⡷⢾⣭⠀⣤⡬⣏⣀⣀⠖⢿⣀⣙⠦⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠃⣚⠋⢘⠃⣾⠆⣼⠇⣿⢁⡚⢀⣡⡿⢀⡖⢸⡟⢰⡿⢠⡟⢠⡟⣰⠆⣿⠁⡣⡮⢶
⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⠁⠇⢾⣦⣵⣤⡵⣦⡴⠃⠑⠬⠭⠦⠞⠣⠬⠲⠬⣴⣬⢥⣬⣴⣿⠬⠕⠤⠭⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠒⠁⠀⠧⠚⠙⢿⣿⣷⣼⢻⣷⡀⠀⠰⡢⢔⠆⠀⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣬⣻⣦⡈⠠⠄⣠⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣍⢻⣶⣾⡿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠹⡵⠁⠀⠀⠀⡀⠐⠀⢀⣀⣀⣀⠀⠀⠂⢄⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡶⠄⠀⠚
⠀⢠⠀⠀⢸⣀⡠⠐⣒⢒⡒⠶⠀⠀⠀⠀⠀⢀⡎⠀⠀⢶⣟⠁⠉⠉⣉⣷⠀⠈⢃⠀⠈⢧⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⢘⡀⠀⠸⡇⠰⠈⠁⠀⠉⠁⠠⢀⣸⠀⠀⠸⣟⣆⡀⠀⠃⠀⠀⠀⢼⡷⠲⢶⣿⠀⠀⠀⠙⠛⠦⢤⣀⣀⣄⣀⣤⠴⠃
⠀⠈⠳⠆⠀⠈⠁⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠈⠻⠟⠛⠳⣤⢄⣄⣸⣇
⠀⠀⠀⠀⣀⠥⠄⠾⠷⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⠿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠄⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠈⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠑⠀⡄⠀⠀⠀⠀⡀⡼⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢠⠀⠐⠂⠀⡀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢸⠆⠀⠁⠀⠘⠛⡇⠀⠀⢀⠁⠀⠀⠀⢀⡉⢠⣠⠤⠠⢄⣤⠀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠡⠀⠀⣆⡀⢀⡌⠁⠀⢀⠀⠡⡄⢀⣼⠃
⠀⠀⢸⡆⠀⠀⠁⠄⠀⢀⠀⠈⠁⠀⢀⡄⠀⠈⠷⣄⣀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠞⠀⠀⠀⠹⣧⣀⣀⡜⠀⠈⠀⣀⢀⣾⠟
⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠁⠀⠀⠀⠈⠉⠙⠛⠲⠦⠴⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠛⠚⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠨⠤⠤⠦⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⣠⣶⡭⠽⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⡄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠙⠦⣄⠀⠋⠁⠀⠀⠈⢹⡇
⠀⠀⠀⠀⢠⠄⠀⠀⠀⢰⠀⠀⡍⠓⠂⠀⠀⡆⠀⠀⠹⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⡠⠀⠀⠀⠘⠷⢤⣀⠉⠛⠛⠋⣀⠔⠁
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠃⠀⠀⠤⠤⠀⠀⠀⠀⠀⠈⠉⠨⠀⠋⠁
⠀⠀⠀⠱⢄⣀⣀⡀⢀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠘⠀⠢⠦⠀⠄⢀⣠⡤⠖⠉⠲⠤⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠰⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣶⣦⣾⣿⣿⣿⣷⣤⣦⣤⣤⣠⠂⠀⠀⠀⠀⢀⡄⠀⠀⣷
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⡀⣀⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠤⠶⠾⠿⠿⢿⠟⠀⠀⠀⣟⢀⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⢶⣾⣿⣿⣿⣿⡇
⠀⣦⠄⢀⣀⣀⣀⠀⠠⣶⡦⣤⣤⣀⣀⣀⣀⣀⡐⠒⠛⠛⣏⠈⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⢻⣿⣿⡿⠄⠀⠀⠀⠀⢤⢰⡆⣸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠞⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣿⡿⠓⠠⠾⠐⣿⡇
⠀⣿⠉⢻⣿⣿⣿⣿⣿⣿⠓⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⠀⠀⠤⠶⢾⡇
⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣟⣛⣻⣿⣿⡿⠿⣿⡿⠿⢿⡿⠿⢿⡿⢿⡿⠿⠿⣿⠿⣿⣿⣿⣿⠆⠀⠀⠀⠰⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠉⠛⠃⠀⠠⠭⠅⠈⢩⡟⠉⠉⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⠁⠀⠀⠀⠀⡀⠀⢀⠄⠀⠀⠀⠀⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠄⡄⢀⣄⣠⣏⣸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⠀⠀⠀⠀⠀⢺⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣦⣤⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⡀⢨⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⢀⣄⣀⣿⣟⠛⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠳⠖⠀⠀⣈⣯⣌⡙⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣧⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣯⣤⣌⣉⡉⠛⠛⠛⠛⠛⠻⠿⠷⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣭⣽⣿⣛⣿⣿⣿⡟⠶⠶⠶⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠬⢹⠟⠁⠀⢘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡖⢨⢀⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠁⠘⠈⠉⠙⠛⠒⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀⠤⠀⠀⣂⣶⣀⣀⣀⣤⣿⣤⣤
⠀⠀⠀⣀⣠⡄⠀⠀⠈⠀⠀⠀⠀⣉⣉⠉⣩⡉⣩⠍⢉⢓⠶⠀⢀⡆⠀⢠⡀⠄⣀⠀⠀⠀⠠⠐⢶⣾⣟⢑⡋⢹⠋⢻⣿⡿⠉⣽⠉⢻⡷
⠀⠀⡞⢻⠟⠿⠟⡀⢀⠀⠀⣄⠀⣾⠀⣸⣯⡧⣿⠁⠀⠈⢀⠀⠚⢷⠀⢸⡇⠀⣿⠘⣻⡛⠁⢹⣾⠺⠁⠀⠀⣸⠀⢸⣿⣳⢠⣸⡂⣨⣷⣀
⠀⠀⠁⢰⠀⠀⠀⠁⠘⢸⣷⣿⣿⡯⠀⡅⢿⣷⣼⡇⣶⢀⣸⣀⣴⡞⠀⣼⡇⢠⠇⠀⣽⡇⢠⣾⣻⢶⣇⡒⠀⢾⣦⡞⣽⣿⠋⢸⣏⡿⣿⣿⡄
⠀⠀⠀⠀⠀⢘⣒⣀⣠⣿⣿⣿⣿⣼⣷⣛⡿⣿⣼⣷⣯⣿⣷⣷⣬⣥⣷⣟⣹⣶⣷⣾⣶⣧⣾⣶⣶⣾⣿⣿⣷⣾⣿⣥⣤⣏⣔⣿⣿⣷⣷⣿⠁
⠀⠀⠲⠖⠪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣳⣿⣿⡍⢻⢋⠉⠹⠛⠈⠛⠉⠹⢉⣉⣸⠫⢹⡟⠋⠹⢿⣿⠟⠛⡿⢻⠿⣿⢿⣿⣿
⠀⠤⠤⠦⠤⠾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠤⠶⠦⠆⣆⠀⠚⠈⢓⣦⢭⣿⠀⢸⡇⠀⠀⠘⠁⠁⢛⡁⠀⢸⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣟⣻⣿⣿⣹⠛⠋⠻⠏⠀⠀⠀⢀⣤⢹⡇⢈⣻⡄⣤⣤⣽⣧⣬⣧⠴⠀⣀⣤⣰⣾⣷⣤⣽⣃⣤⣦⣀
⠀⠀⠀⢀⡤⣤⠀⠀⠀⠀⠈⠙⠻⠏⠁⢿⣿⣿⣷⡄⢓⢲⣃⣀⣖⣳⠾⠿⠦⠴⢶⣟⣃⣒⣈⣯⣵⣼⣿⣯⣶⣶⣶⣾⣿⡾⠁⠈⠙⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠛⠛⣛⣿⣿⣿⣿⣿⡁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠤⠶⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠏⠉⠉⠁⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⡿⠿⣿⣿⠿⠿⠿⢿⣿⣿⠿⠿⠿⢿⣿⣿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣾⡿⢋⣘⣻⣿⣿⣟⣛⣛⣺⣿⣿⣛⣛⣛⣿⣿⣿⣛⣋⡹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠬⠭⢽⣿⣿⡧⠭⠭⢼⣿⣿⠭⠭⠭⣿⣿⣿⠬⠭⠥⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⢓⣒⣺⣿⣿⣗⣒⣒⢺⣿⣿⣒⣒⣓⣿⣿⣿⣚⣒⡒⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣌⠩⢽⣿⣿⡯⠭⠭⢼⣿⣿⠭⠭⠭⣿⣿⣿⠭⢍⣱⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣿⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⡿⡶⡿⡿⠛⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠯⠼⠦⠶⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⠆⢦⣤⣦⠐⠿⠿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⡸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡠⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢉⣹⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣄⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣭⣶⣤⣤⣆⣀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠂⢹⣏⣽⣶⠀⠀⠀⠠⠄⠀⠀⠠⠤⢆⣀⣴⣿⠱⢟⠋
⠀⠀⠀⠀⣀⣀⡀⠀⠠⠄⠀⠐⢟⣻⢿⣿⣿⣿⣿⣶⣶⣞⣀⡀⠂⡀⠀⠀⢠⣼⣿⣿⡟⠀⠀⣀⣀⡠⠦⠐⠒⠉⢹⣿⣿⡇⢸⡿⠂⠀⠐⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠶⠆⠀⢚⣒⡀⣈⢯⣟⠶⠭⢍⣟⣟⣿⣯⣿⣿⣿⣷⣶⣴⣿⣿⣟⠛⢉⣁⣁⢠⡤⠴⡖⠲⠋⡀⠈⢿⡟⠀⠸⡁⠀⠀⠀⠲⠀⢠⣤⠉
⠀⠀⠀⠀⠀⠀⣀⣂⣀⣂⣤⣀⣀⣈⡻⣿⣿⣶⡶⢶⣤⣥⣉⣉⣙⣉⣉⡉⠉⠉⣍⣉⣉⣠⠠⠦⠞⠃⣬⣑⣒⣫⣩⣿⡅⠂⠀⢇⠀⠁⠶⠶⠆
⠀⠦⠤⠀⠤⠦⠤⠴⠆⠁⠀⠀⠉⠩⠽⠿⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣼⣧⣿⣿⣶⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣧⡘⠅⠀⣼⣀⣉⡀⠀⠈⠉⠉⠁
⠀⠀⠀⠐⠚⠛⣛⣛⠋⠙⠛⠛⣿⣶⣆⣀⣀⠀⠉⠉⠙⠛⠻⠻⠟⠛⠿⢿⣿⠿⢿⣿⣿⣿⣟⣻⣿⠿⠛⠉⠁⠠⠄⣀⠍⠒⠀⠻⠒⠶⠤⠄⠀⠉⠒⠠⠤
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠩⠭⠉⠉⠁⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠈⠓⠲⠶⠅⠚⠋⠙⠃⠙⠂⠀⢀⡀⣀⣲⠦⠄⠰⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⡄⠀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠖⠒⠒⠒⠊⠉⠉⠒⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡤⠤⠾⠖⠒⢚⡙⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠙⠻⠶⣤⣀
⢀⠤⠤⠴⠶⠒⠒⠙⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⠐⡖⠒⠒⢤⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤
⡆⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⢠⠀⣴⠴⠶⠶⠆⢶⣄⠀⢸⡀⠀⠘⡇⠀⢸⡄⠀⠸⡄⢻⣠⠴⠚⠀⢸⡤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢹⢾⡀⠀⠀⢰⣆⠀⠀⠀⣏⠙⢦⠘⣆⠀⡌⠀⢻⣀⣀⣀⠀⢸⡍⢷⡈⣇⠀⠀⢧⠀⠈⢧⡀⢸⠃⢸⡟⢦⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠘⣿⣇⠀⠀⢸⠘⣆⠀⠀⢹⡀⢘⡇⠹⡄⡇⠀⢸⡇⠀⠀⠀⠈⡇⠀⠙⢿⠀⠀⠸⠄⠀⠈⠙⠃⠀⠀⠃⠀⠙⠂⠀⠉⠉⠉⠁⠀⠀⠀⡠⠖⠁
⠀⢿⣿⠀⠀⡷⠖⠘⣆⠀⠈⣇⠜⠁⠀⢻⠁⠀⠀⠷⠶⠶⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠊
⠀⢸⣿⡀⠀⠃⠀⠀⠙⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢀⠀⠠⢤⣤⣤⣤⣶⣶⣦⣤⣤⣴⠶⠶⠶⠶⠟⠛⠋
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡤⠤⣤⣴⣖⣲⠶⣶⠶⠶⠖⠻⠟⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⢙⣳⣾⣤⣐⠶⠶⠮⠿⠛⠓⠛⠊⠉⠉⠁⠀⠀⠀⠀⢸⡆⠀⠀⠀⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠳⠀⠀⠀⠸⣤⡆⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠐⡄⢠⠀⠀⠀⢀⡿⠀⠋⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⢆⠘⢆⢇⠀⠀⣼⣷⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠣⠘⡞⠀⡈⣿⢿⣀⠞⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡉⠪⠐⠄⢹⠷⣗⣨⠁⣁⠀⠀⠲⠊⠁⠀⠀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠈⠁⠠⠤⠭⠤⠄⠀⠐⠟⠛⠒⠚⠛⠒⠒⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠠⠁⠀⠀⠉
⢶⡐⠒⠒⠒⠒⠀⠀⠀⠤⠤⠤⠤⠀⠀⠠⠤⠤⡤⠀⠀⠀⠀⠈⠈⠉⢳⡂⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⣀⢆
⠀⠪⠑⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠘⠇⠨⠭⠭⠽⠁⠀⠀⠙⣦⠤⠤⠄⣀⡀⠀⠀⠀⠈⢂⠀⠀⠑⠄
⠀⠀⠐⢤⣀⡀⠉⠀⠒⠒⠠⠤⠤⢤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⢁⢀⣀⣀⢁⠃⠀⠈⠀⠢⡉⠢⡀⠀⠀⠀⠉⠉⠉⠙⠂
⠀⠀⠀⠀⠙⠶⡩⠌⠃⢒⣂⠠⠤⠤⠄⠀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠛⠓⠒⠒⠒⠂⠤⠤⠤⠬⣭⣀⣘⣀⡐⠒⠚⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⢀⣀⠉⠉⠀⠐⠒⠒⠒⠋⠉⠭⠥⠀⠠⣤⣄⣒⣒⣒⣒⣒⣂⣀⠤⠤⠤⠤⠤⠬⠭⠭⠉⠿⠶⠿⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠉⠁⠐⠀⠒⠂⠠⠤⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢉⡅⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠠⠤⠄⠀⢤⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠉⠁
⣿⣿⡷⠷⠿⠾⣿⣿⣿⣿⣿⣶⣿⣾⣷⣾⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣰⣿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠃⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⢸⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢛⠛⠛⠃⠿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣠⣄⣀⡀⠀⢸⣷⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⣿⠇⣿⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⡚⠃⣠⡤⠀⠀⠃⠀⢸⡄⠀⠀⠀⡏⠀⠀⢤⠀⠀⠀⠀⠀⠈⢹⣿⣿⡀⣿⡇
⠀⠀⠀⠠⢰⣷⢶⣆⣤⣠⡄⢤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣁⠐⡧⠀⠀⠤⢸⠤⠤⠤⠤⡧⠤⠤⠤⠤⠶⠤⠦⠤⣶⣾⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⢿⣷⡟⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⡿⠹⠟⠁⠀⠀⠀⠀⠀⣹⠀⣴⡄⠀⣄⠀⠀⠀⠀⣿⣿⣴⣿⣾⣿⣟⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠋⠁⠲⠶⠅⠀⡀⠀⠀⠀⠀⠀⣀⠀⡾⢀⡙⠃⣸⣿⡷⠤⠤⠤⢿⣿⣿⣿⣿⣿⡇⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣾⡿⠖⠒⣂⣭⣥⠇⣸⣷⡿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⣀⣠⡴⠒⢛⡋⠍⠚⠈⢃⣠⡶⠿⡿⢿⣷⠿⠋⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣏⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢾⣿⣿⡏⣯⣥⢖⢦⣘⣀⣠⣴⠞⠛⣱⣶⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⡇
⠀⠀⠀⣠⡴⠶⠛⠉⣃⠠⢸⣿⣿⢠⡿⣷⣶⣶⣶⣶⣶⣖⡤⠄⣾⣿⣿⠿⠤⠤⠤⠤⢤⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⡇
⠀⠀⠀⢀⣠⠤⠒⠉⣏⢀⣰⣿⡷⠚⢋⣉⠉⠉⣉⣭⣭⣭⣵⡿⢿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣐⣒⣒⣒⣒⣲⣾⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠃⠠⣠⣴⣮⣿⠉⡴⣿⣥⡴⠛⣋⣴⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠶⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⣛⣭⣿⣿⣿⣿⡿⠋⣠⣴⣾⣿⣿⣿⡿⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡿⠟⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡿⠟⢋⣤⣶⣿⣿⣿⣿⣿⡿⠟⠡⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢋⡐⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀

⠀⠀⠀⠰⡀
⠀⠀⢈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⣄⡀
⠀⠀⡉⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠡⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠈⠃⠀⠀⠀⠀⢀⣠⠴⠦⢄⡀⠀⠀⠀⠀⠠⢀⠨⠀⠀⠀⠀⠃⠂⠀⠀⠀⣤⡀⣤⣀⡀
⠀⠀⠐⠀⠀⡷⠐⠀⠀⠀⠉⠀⢀⠀⠀⠄⠀⡀⠀⠀⠀⠐⡂⠀⠀⠐⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢡⠦⡀⠀⠄⠀⠀⠀⢤⡈⠉⠠⠀⠀⡀⠀⠄⠀⠀⠁⠃⣴⠆⣀⠀⠀⢶⠀⠀⠀⠰⠄⠀⠀⢠
⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠙⠀⠀⠀⡄⠀⠀⠀⣀⠀⠀⢐⠀⠀⠁⢀⢀⣠⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⡧⠠⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⡀⠀⠀⠄⠀⠀⠀⠃
⠀⠀⠑⠋⠀⠀⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣤⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣧⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⣿⠀⠘⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣻⣆⡄⠀⠀⣿⡆⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠅⠀⠀⠿⠃⠀⠀⠀⠀⠱⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣤⣠⡅⢀⣤⠀⠰⣀⣀⣽⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠟⣿⣿⢿⣯⣷⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠂⢈⣀⣠⣶⣿⣧⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⡼⠛⢰⣛⡩⠠⣶⣫⣽⣿⣿⣽⡿⠟⠉⠁⠀⠀⣼⣻⣿⣶⣯⡛⠓
⠀⠀⠀⠀⠀⠀⣀⣼⣧⣠⣼⣷⣾⣿⣿⠿⠟⠋⠉⠀⠀⠀⢄⠀⠃⠋⢨⡾⠛⠛⠃⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡙⣟⣛⠛⠋⠉⠉⠁⠀⣠⣠⣶⣶⣶⣶⡿⠿⠀⠀⣶⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⡟⢁⣴⣶⣿⢟⡕⠙⣽⡿⠿⠿⠧⠀⢺⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠿⠮⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⠀⢸⣏⣟⣽⣿⣽⣿⣽⣿⣯⣿⣫⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⣦⠸⡦⠀⠀⠃⠚⠃⠀⠰⣉⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠢⡼⠈⢀⣀⣀⠀⠀⠐⠂⠀⠀⠠⠀⠀⠓⠃⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡙⠂⡀⠀⡏⡏⠀⢀⠀⠀⠀⢺⠆⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠃⠀⡀⠀⠀⠑⠬⠶⠋⠀⣀⡀⠀⠀⠀⠈⠀⣴⠄⠀⠀⢁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢍⠛⠁⠀⠟⢀⣤⠀⠀⠀⠉⣰⢻⢳⡠⠖⡆⠀⠀⢈⡀⠈⠁⠀⢚⣁
⠀⠀⠀⠀⠀⠀⠀⢐⣊⠁⢀⠀⡠⠋⡏⢧⠐⠀⡰⠃⡜⠀⢁⡴⠁⠂⣠⠚⡟⣆⠀⡀⠙⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠸⠴⠃⠀⠚⠁⠘⣅⡸⠳⡞⠀⠸⠵⠂⡞⣞⠱⠘⠹⣤⠇⢈⠢⠀⠐⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠺⠄⠀⢼⡧⠀⠈⠁⠀⠘⠄⠀⠀⠀⠘⣌⡆⠑⠀⠈⠀⢾⠦⠀⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣺⣟⠀⢠⣿⡄⠀⢰⠒⠦⠤⢦⡈⠇⣰⣷⡀⠐⣿⣗⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⢽⡿⡁⠰⣿⡢⢠⠋⠓⠤⢎⢿⣱⡄⢞⣿⠆⠨⢾⡮⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣻⣏⡂⣪⣿⢝⠀⠭⠍⠑⠚⠉⠥⠀⡻⡿⡅⢐⣽⣝⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠉⢔⣿⡳⠄⣱⣆⢩⠍⣴⡌⠠⢚⣟⡢⠉⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⠌⠂⢽⡷⡀⠨⣾⢮⠘⠵⡷⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣶⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⣀⠀⢠⡄
⠀⠀⠀⠀⠐⠤⣤⣭⣅⠒⠒⠠⠤⠄⠠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠈⢢⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⣀⣀⣀⣀⣩⢰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡟⠀⣿⣿⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⡤⣤⠤⠤⠤⠤⡤
⠀⠀⠀⠀⠀⢀⣼⡿⠀⣿⠀⣴⡿⠻⣶⣿⣶⣶⣿⠶⠠⡜⠀⠀⠀⠊⠀⠀⠀⠊⠀⠀⠀⠜⠉⠉⠉⢑⣛⡂⣤
⠀⠀⠀⡤⠴⠚⣛⣛⡛⠛⡻⠛⠧⣤⡤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠐⣒⣶⠂⠀⠀⣭⡭⠉⢉⣉⣀⣠⣤⠖⠁
⠀⠀⡞⠀⠀⠀⢀⣀⠀⠀⠀⠉⠭⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠁⣀⣤⠞⠁
⠘⠫⠥⠤⠾⠭⠤⠤⠤⠤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢔⡅⠄⠒⠒⣠⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠄⢀⣴⠋⠤⠤⠴⠶⣭⣁⣀⣀⣀⣈⣉⡉⠁⠉⠉⣉⣉⣐⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠄⡾⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⠾⠿⠟⠛⠋⡩⠏
⠀⠀⠀⠀⢀⡤⠂⠐⠁⠀⠀⢀⣳⣤⣴⣶⣾⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⣀⣤⡶⠋
⠀⠀⣠⣴⣯⣤⣤⣴⠶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠢⠄⠈⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⡸⣿⡅⠀⠐⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠴⠠⠂⠈⠉⢀⠀⠀⠚⠃⠐⠀⠀⠢⠒⠂⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡤⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⢨⠟⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣻⣤⣀⣀⣂⣀⣀⣀⣀
⣿⣿⠛⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⢹⡀⠀⠀⠈⢺⠉⠉⠐⠒⠢⡆⠤⢄⣀⣴⢁⠁⠘⠓⠌⠙⡀⠀⠈⢣⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⢘⡥⢆⡠⠀⠀⠷⠤⠤⠤⡤⠴⠄⢠⣮⠤⢜⡫⣶⣒⡿⣆⣠⡈⠲⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⡥⢻⠀⠀⣰⠃
⣎⠃⠸⣏⠉⠁⢠⣠⣀⣊⡀⠐⠂⠨⢅⣙⠛⠛⢮⡻⢿⠘⣿⣿⣤⣽⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⠁⠁⢰⣾⠏
⠀⠙⠶⡬⢤⣁⡚⠿⠍⠻⠿⢿⣷⣦⣀⡀⠈⠑⠚⠛⠷⡀⠈⣁⣿⣿⣿⠿⡸⠿⠁⠓⢤⣀⡀⠘⡀⠴⠨⡟
⠀⠀⠀⠈⠢⡀⠉⠙⠓⠲⠤⢥⣀⣻⡠⠈⠉⠐⠒⠤⡀⠀⠀⠈⢏⠉⠁⠎⣀⡞⠀⠀⠘⢠⠄⠈⢹⣶⣶⣗⠠⠄⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠈⢶⣄⡀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠿⢷⣿⣶⣦⣤⣈⣻⣯⣾⡛⢣⡴⢶⠀⢸⠀⠀⠘⣿⣿⡿⡇⠂⢠⣾⠇⠤⢩⣴⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠷⠶⣤⣭⣉⣛⠛⠛⠋⠁⢖⡋⢋⠇⠛⠒⢺⡇⣀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⢒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⠦⢤⣬⣁⣐⡛⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠭⠽⣛⣲⣶⢦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠶⢶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⢠⠿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠭⠿⢓⡲⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠤⠤⠄⠀⠀⠀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡄⠀⢸⠀⠀⣠⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢤⠀⠀⢳⠀⠀⢷⠀⢸⠀⠀⡏⠀⢰⠇⠀⣠⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢧⡀⠈⢧⠀⢸⠀⢸⠀⢰⠇⢀⡏⠀⡴⠁⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠦⣀⠀⠈⢦⡀⠳⡀⡘⡆⡞⣧⣽⢸⣸⢰⣼⣄⡼⡁⣠⠞⠁⢀⡴⠂⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠈⠳⢤⡀⡙⢿⣽⣴⣹⣄⣿⣾⣠⣿⣹⣿⣞⢞⢜⠅⣠⠞⠁⢀⣠⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠦⣀⠽⣺⣷⣽⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⢟⡵⡊⣕⡤⠖⠉⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣐⡪⢽⣺⣽⣿⣿⣿⣿⠿⠛⠿⠿⣿⣿⣿⣿⡿⠏⣓⣪⠤⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⠤⠤⢤⣄⣀⣐⣾⣯⣝⣺⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣯⣕⣒⣶⡤⠤⠤⠤⠴⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠻⢭⣭⣻⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡶⣶⠋⠉⡓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠵⠚⠭⠀⠀⠀⠀⠀⠈⠙⠻⠿⢤⣀⣀⣀⠀⣠⠴⢚⡿⠟⠋⠁⠀⠄⠀⠀⡠⠉⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⠁⠀⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠂⠈⠉⠀⠈⠙⠦⢤⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠨⠥⠤⠤⠤⠤⠤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠛⣿⢸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⢀⣿⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⢶⡇⣿⢸⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⠇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣋⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⠉⠉⠉⢹⡆⣿⢸⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡿⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⢸⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠈⢧⡀
⠀⠀⠀⠀⢠⡏⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⢸⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⣼⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⣀⡤⠶⠛⠒⠛⠓⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠶⠦⠤⣤⣀⣀⣀⠀⠀⣀⣤⣚⣁⣀⣀
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⢀⡼
⢯⣤⣤⣤⠤⠤⠤⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋

⠀⠀⢴⣶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⢀⣼⣷⢶⠖⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣦⠀⣨⠿⣄⠀⠀⠀⠀⠀⢠⣬⡷⠞⠁⣿⠀⠀⠀⠀⠀⠀⢠⣶⣬⡉
⠀⢸⣝⣿⣧⢶⣄⠀⣼⡇⠀⠀⠀⠀⠿⢷⣼⠿⣥⡀⠈⠀⠀⠀⠀⠀⣾⣏⡛⠋⣼⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠶
⠀⠜⠁⠀⠻⣆⠈⣴⣿⡇⠀⠀⠀⠀⣠⠞⠉⢿⣬⣿⡄⠀⠀⠀⠀⠀⢸⡏⠻⣠⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⢻⡄⡀
⠀⠘⠀⠀⠀⢉⣼⣿⣿⡇⠀⠀⠀⠈⠁⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⢘⣣⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃⠁

⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢄⡀
⠀⠀⠀⠀⣴⣷⣤⣄⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⡠⠊⠁⠀⠀⠀⠀⠀⢘⣛⠀⠀⣛⡃⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠐⠠
⠀⢐⡶⣤⣿⣷⣴⡾⠷⠄⠀⠀⠀⠀⠀⣀⡸⠫⣄⣀⠀⠀⠀⠀⣠⣴⣿⣯⣷⣼⣿⣽⣦⡄⠀⠀⠀⠈⢛⣧⡴⠟
⣼⣟⣻⡯⠿⠿⠏⣾⠿⣷⡄⠀⠀⣤⣾⡿⠃⠀⠻⣿⣷⡄⠀⠀⢎⣭⡝⠭⢭⣭⠯⠯⠭⠅⠀⠀⠀⠀⠌⠻⣷⣶⣦⣄
⠈⠛⠛⠁⠀⠀⠀⠛⠿⠛⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⠋⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠉

⠀⠀⠀⢀⠠⡠⡄⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀
⣴⣔⣊⣹⣞⣓⣼⣦⣤⣄⠀⠀⠀⠉⠱⠀⢸⡧⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⡀
⠻⣿⠿⣿⣿⣿⣿⣿⠿⣿⠆⠀⠀⠀⠀⢧⣼⣷⣿⣿⣶⡄⠀⠀⠀⠀⣠⣿⣿⠹⣷⡀⠀⠀⠀⠀⣶⣀⣀⣠⣿⣻⣃⣀⣠
⠀⠻⠶⠟⣁⣀⣈⠻⠶⠟⠀⠀⠀⠀⠀⣿⣿⠉⠀⠈⠉⠀⠀⠀⣀⣼⣿⣿⣿⠀⠙⣿⣄⡀⠀⠀⢙⡛⣛⣛⡛⣛⣛⡛⣛
⠤⠶⠿⠿⠿⠿⠿⠿⠶⠦⠀⠀⠀⠀⠤⠴⠶⠶⠶⠶⠤⠄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⢺⠂⠀⣠⣶⠄⠀⠀⠀⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⢠
⢀⣀⣤⣤⣿⣷⣾⣭⣄⣀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣉⠳⠂⠀⠀⠀⠀⣀⣀⣼⣿⣲⣴⣷⣦⡀
⢨⣍⣩⠭⣭⣩⡭⣭⣩⡥⠀⠀⠀⠀⠀⠀⠀⠶⢼⡃⠀⠀⠀⠀⡾⠉⠙⣮⢿⣉⡞⠉⠙⡆⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⡄⠀⠀⠻⠤⠴⠋⠈⠁⠳⠤⠴⠃⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉