Cup symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⣼⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣸⣿⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠤⠴⣿⠥⠤⣾⢯⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡩⢭⣶⣶⡶⠒⠊⠉⠉⠙⢏⠉⠻⠓⠒⠒⠒⠨⠭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣧⣜⣻⠿⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢨⣍⠙⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠉⠁⢀⣿⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡴⠶⠶⣌⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡟⠀⠀⠀⠘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣸⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⢾⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣱⠯⣅⣀⣀
⠀⠀⠀⣀⡤⣖⣺⠭⠗⠚⠉⠉⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢧⠀⠀⠐⠚⠭⢭⣒⡢⢤⣀
⠀⢠⡞⣵⠟⠉⠀⠀⢀⣀⠤⠤⣒⣒⠚⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣬⠷⠤⢤⣀⠀⠀⠈⠙⠳⣮⢳⡄
⠀⠸⡎⣿⠀⠀⢠⠖⣩⣶⡾⠋⠉⠀⠀⣿⡸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⠙⠲⣦⣍⠲⡄⠀⠀⠘⣰⠃
⠀⠀⠙⢮⡳⣄⠸⣄⠻⣿⣦⣀⠀⠀⠀⠈⠷⣬⣓⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠈⠿⣣⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠉⠒⠭⣂⡉⠒⠪⠭⢟⣒⣲⣤⠤⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⣀⣀⡀⠤⠴⠒⣉⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⣄⣉⣉⣉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⢋⣉⣉⣉⡥⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠒⠛⠛⠛⠉⠉⠉⠉⠛⠛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠲⠤⣤⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣠⡤⠶⠚⠉⢸⣤⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠳⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡴⠋⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠲⠶⠶⠶⠶⠶⠖⠒⠚⠋⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠘⣦⡀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣤⣴⣾⣿⡟⠀⠀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⠿⠋⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⡿⠟⠉⢉⣁⣀⣀⣀⣀⣀⣈⡉⠉⠛⠻⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠉⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣤⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⣀⠉⠉⠉⠉⠉⠉⠉⠉⢉⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⠃⠛⠁⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣀⣤⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣾⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣿⣿⣿⡏
⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠛⠻⠛⠁⣀⡀
⠀⣠⣴⣿⡿⠟⢋⣩⣤⣶⣶⣿⣷⣦⡀⠈⠙⠻⠿⢿⣿⣿⡿⠿⠟⠛⠁⢀⣴⣾⣿⣶⣶⣤⣄⡚⠿⣿⣷⣤⡀
⣼⣿⡿⢉⣴⣾⣿⣿⣿⣿⡿⠋⠉⠀⣀⣤⣶⣤⣤⣀⣀⣀⣀⣠⣤⣦⣤⡀⠀⠉⠛⢿⣿⣿⣿⣿⣷⣄⠙⣿⣿⡆
⢿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇
⠈⠛⢷⣌⠻⢿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣈⡉⠙⠛⠛⠛⠛⠛⠛⠋⣉⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⢋⣴⠿⠋
⠀⠀⠀⠈⠛⠲⢮⣝⣛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣭⠵⠚⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣓⣒⣚⣋⣉⣉⣭⣭⣭⣭⣭⣭⣭⣍⣉⣉⣛⣒⣀⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣀⣀⣀⣀⣀⣀⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡆⣶⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⢹⡏⠹⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠸⣿⢸⠏⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣭⣧⣯⡄⣿⣧⡀⣿⣰⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆⢶⣶⣶⣾⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⢀⣾⣿⠃⠀⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⢀⣾⣿⠋⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣱⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⣰⡿⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠁⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⢿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣾⣿⣿⣷⣶⣦⢠⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⣿⣿⣿⣿⠟⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⢹⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣧⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠺⣷⡄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢸⣿⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⢀⣾⠟⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⣰⡿⠃⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠄⠙⣿⡄⠀⠻⣧
⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄
⠀⠀⠀⠀⢽⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣇⣴⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠀⠈⠙⣿⣷
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢿⣿⣤⣀⣀⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠙⠻⠿⠿⠟⠋⠁
⠀⢠⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶
⠀⠀⠙⢿⣿⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⡿⠋
⠀⠀⠀⠈⠛⢿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⢴⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡠⠴⠒⠌⠁⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⣄
⠀⠀⠀⠀⠀⠸⠧⣄⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⢀⣀⡡⢾
⠀⠀⠀⠀⠀⠈⡇⠀⠈⠉⠙⠒⠛⠺⠿⠭⠭⠭⠭⠭⠭⠭⣭⣭⠭⠭⠉⠉⠉⠉⠽⠿⠖⠛⠒⠊⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⠤⠖⠒⠋⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠂⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠋⠀⠀⣀⠤⠤⠤⣄⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡠⠖⠉⠀⠀⠀⠀⢈⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⢀⡠⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣠⠤⠖⠉⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠁⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠤⠔⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠒⠚⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡹⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡞⠛⠛⠛⠛⠛⠓⠛⠛⠓⠛⠛⠛⠛⠛⣦⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢩⣭⣭⣭⣭⣬⣭⣭⣬⣭⣬⣭⣭⣭⣥⠹⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⢹⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣎⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡏⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⠖⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⢟⣫⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣀⣀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣛⠿⠿⠿⠿⠿⠟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⣄⣀⣈⣷⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠛⠛⠉⠉⠉⠁⢸⠊⠀⣀⣀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠱⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠊⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣀⠤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠀⠒⠂⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⠦⠤⠤⣀⣀
⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠤⠬⠍⠉⣛⡒⠒⠒⠒⠛⠛⠭⠍⠉⠤⠤⠀⠐⠂⠀⠁⠀⠀⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠈⠻⢍⡑⠒⠢⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⢚⣩⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣄⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣠⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⠼⣷⠤⠼⣧⠤⠤⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠁⠀⠀⠀⠀⢀⣈⣀⣀⣛⣀⣀⡀⠀⠀⠈⠉⠙⠻⣦⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣶⣚⣛⣋⣉⣉⣉⣉⣉⣉⣉⣉⣍⣉⣉⣛⣻⣶⠶⠴⢺⣇⣀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡿⠀⠀⠀⠀⠀⠀⢸⠏⠉⠛⢶⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣸⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢛⣶⣄⠀⠀⠀⠀⢠⠟⠶⠖⢊⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢠⡺⡇⢻⡀⠀⠀⢠⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿⢻⠕⠀⣘⡇⢀⠔⠉⠙⠛⠒⠦⢤⣀
⠀⠀⠀⠀⢸⡉⠀⠀⠀⠀⠀⠀⢼⣿⣷⣦⣄⣀⠀⠀⠙⠛⢛⣋⣥⣶⣿⣦⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠙⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁⠀⠀⢀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣶⣶⣤⡤⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣶⣶⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠚⠉⠙⠩⠭⠉⡉⠉⠁⠤⠈⠀⠀⠈⠉⠉⠩⠍⠉⠉⠙⠒⠒⠒⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⠥⠤⠒⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢵⣿⣿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣙⠛⠲⠦⠤⠤⠤⣤⣄⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣥⠤⠤⠴⠶⠞⠛⣉⣴⣶⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠛⠶⠤⠤⠶⠤⠄⠈⣉⣉⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⢈⣉⠉⠀⠤⠴⠤⠤⠴⠞⠋⢩⣿⣿⡁⠤⠀⠀⠉⢧
⢀⡄⡂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡴⢲⡆⠀⠀⣠⡄⠀⠀⠀⣉⠉⠉⠉⢉⡉⢉⡁⠀⠠⠀⠀⠀⠀⠀⠀⣴⢶⠀⠀⣿⣼⢻⢧⠀⠘⠓⠀⡼
⠘⢦⣥⠄⠲⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠷⢬⣠⡀⠀⠙⠂⢠⠟⠛⢿⣷⡄⠀⣨⠟⠄⠀⠀⢶⣦⣀⣠⣿⣿⢶⡔⠁⠀⡜⠙⠶⠿⠋⢠⠃⠀⣰⠃
⠀⠀⠀⠉⠐⣄⠈⠀⠡⡀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠁⠈⣠⡴⢦⡀⠘⠄⠇⠀⣿⠁⠊⣤⣶⡄⠓⠀⢩⢿⣯⡉⠉⠃⠀⠄⠀⣼⠀⢀⣀⠠⠚⢁⣤⠖⠁
⠀⠀⠀⠀⠀⠈⠛⠻⢦⣌⠂⣠⢂⠥⠂⠁⡼⣷⡴⠀⠰⣏⣰⠀⣹⣤⡆⢀⡼⠑⠒⢠⣧⣁⣶⡤⣾⣁⣄⠛⠁⣼⡦⣄⠀⣸⡵⠟⣉⣡⢴⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡳⣎⡙⠀⠯⠤⠆⡈⢳⠀⠀⠈⠁⠴⠛⣻⣧⢸⠆⣘⣯⡙⡆⢻⣻⠑⡋⣽⣷⠆⢰⣿⣤⠉⣰⠋⣰⣿⡉⢉⣭⡻⢮⡢⢄
⠀⠀⠀⠀⠀⠀⠀⡰⢡⢎⠔⠉⠻⣦⡑⢤⡒⠛⠏⣷⡈⢠⣄⣀⠀⠨⣿⣿⠀⢉⠛⠻⠿⣾⣿⡿⠋⣭⠦⡄⢠⣿⠁⢠⢧⣀⣉⡾⠃⠻⣄⣀⣠⡈⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⣠⣶⡄⠈⢿⣆⠙⢇⠰⠼⣷⣄⠁⠋⢀⣠⡁⠈⠓⢬⣀⢀⡀⠈⣯⣀⠀⡓⢂⣀⠼⠁⢠⠏⢦⡉⠁⢀⣠⣶⡄⢹⣿⡅⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢀⡃⢧⣳⣅⠈⣠⠄⣰⢻⣧⡨⣳⣦⡈⢿⣢⠀⠀⢨⠤⣦⠴⠢⣬⡍⠉⣴⠂⠹⠺⠑⠋⠁⢀⣰⠋⣾⠛⠻⣿⢻⡍⣰⣧⠼⢋⣴⣿⠇⡣
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣮⣿⠧⠤⢯⠄⢿⣿⠈⠛⠛⠓⠛⠿⠶⠦⣄⣉⣶⣂⡻⠀⠀⢾⡅⠶⠀⠤⣣⣾⣟⣁⠄⠉⠀⠀⣿⡘⠿⢃⡞⠰⣫⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⢷⣄⡰⠆⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠳⢦⣤⣤⣭⣤⣶⣶⡾⠿⠿⠛⠃⠀⣠⡤⣤⣌⣉⠋⣉⣴⠿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⡛⠳⣌⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠓⠂⠀⠀⠀⠀⠀⠐⢯⡠⠀⣀⠴⠚⣋⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢮⣝⣿⣷⡶⢤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣠⡧⢀⣀⡀⠀⠀⢀⣀⣠⠤⠄⣒⣭⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠮⠭⢭⣭⣭⣿⣿⣿⣿⣞⣓⣲⣦⠭⠭⠴⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣛⡿⠿⠷⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠤⣭⣉⡓⣶⣄
⠀⠀⠀⠀⠀⠀⠸⣅⠙⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠉⠓⠦⣬⣉⡉⠛⠓⠲⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠶⠒⠋⢉⣠⣾⠃⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣤⡀⠀⠉⠙⠓⠒⠶⠦⢤⣤⣄⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣠⣤⡤⠶⠒⠛⠉⠀⢸⣀⣸⠇⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠰⣿⣿⣿⠂⠀⠰⡷⠀⠀⠀⠀⢀⣀⡀⠈⠉⠉⠉⠉⠉⠉⣡⣤⣄⠀⠀⠀⠀⢸⣿⠀⣿⠀⠈⢷⠀⠈⠋⢹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗⠀⠀⠀⠃⠀⠀⢸⣿⣿⣿⡇⠀⠤⠀⠈⠁⠀⡟⠀⠀⢸⠀⣀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⠀⠀⠈⠀⠀⣠⣤⣄⠀⠀⠁⠀⠀⣀⣄⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⡀⢰⡇⠀⢀⡾⠘⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣿⡗⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠻⠟⠀⠀⢀⣤⡀⠀⠀⢰⣿⣿⠀⠀⠀⣸⠤⠖⠋⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠈⠛⠋⠀⠀⠀⡟⠰⠀⠀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠐⠂⠀⠈⠛⠉⠀⠀⠀⠀⢠⣶⡄⣸⠷⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠉⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠛⠛⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠈⠁⠀⡄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⣀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠙⠓⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠉⣰⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠒⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢹⡆⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣁⣀⣈⣁⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣠⠴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣠⣿⢶⣚⡛⢶⡀⠐⠄
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠈⣿⠉⠛⠛⠿⠾⠿⠿⢿⣿⣿⠿⠿⠿⠿⠟⠛⠛⢹⡟⠋⠉⣹⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣠⡴⣿⢤⡀⠀⠀⠀⠀⠀⣼⣁⡤⠞⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⠀⠋⠀⡇⠀⠀⠀⢀⣼⣯⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣹⣦⡀⠀⠀⠀⢿⣤⣤⣤⡇⠀⠀⣠⣾⣉⡁⠀⠀⠀⠀⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⠉⠀⠀⠙⠳⠶⠶⠶⠶⠦⠤⠶⠶⠛⠁⠀⠈⢉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠲⠶⠶⠶⠶⠤⠤⠤⠴⠶⠶⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣯⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢠⡄⢰⢰⡟⠋⣳⠀⠀⠀⣾⢳⡄⣸⣃⠀⠀⠀⣿⢰⣿⠇⢰⣻⡇
⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠴⠀⣧⠾⠀⣧⡴⠋⠀⠀⠀⠙⠞⠁⢸⡇⠀⠀⠀⠿⠈⠛⠚⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠒⠒⠛⠛⠛⠛⢒⣒⣒⣲⣶⣶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡚⢍⡀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢄⠉⠉⠉⠉⠀⠸⠝⣷⣄
⠀⠀⠀⠀⠀⠀⣸⠹⡶⠿⢿⣿⣛⣻⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣛⣛⣿⣭⡵⠶⠚⠋⢹⣿⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⠨⠅⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⢀⠀⣀⢸⣿⣤⡶⠛⣛⣩⣭⣉⡛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⢒⣒⣠⡭⠭⠄⠐⢸⣿⢇⢤⡞⠉⠁⠀⠈⠙⣦⠈⢿⣦
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣚⣩⠍⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⠀⠸⡇⠈⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⡒⠚⠀⢸⡟⠾⠋⠀⠀⠀⠀⠀⠀⢀⠇⢰⡟
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⡭⠥⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣒⣂⡈⠉⣸⡿⠀⠀⠀⠀⣀⣠⠴⠞⠁⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠤⠤⠆⢠⣿⣃⣤⠶⢚⣋⣩⣤⣦⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣋⠉⢀⣾⣿⡯⣲⣭⡿⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠀⢀⣾⡋⢊⡼⠋⠁
⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠾⣆⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⣋⠅⣾⡟⠛⠛⠷⣶⠶⣶⣤⣤⣀
⠀⣠⠴⠒⠋⠉⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠾⠯⠥⠤⠤⠉⠀⠤⠤⠈⣑⣒⣃⣰⣿⣀⣀⣀⣿⣿⠿⠿⠛⠛⠟⠛⢷⣄⠀⠀⠤
⠞⠑⠀⠀⠀⠀⢠⠞⠽⠇⠀⠘⣷⡀⠀⠀⠤⠬⣯⣭⣍⣩⣤⣤⠭⠽⠿⠛⣛⣩⣾⡿⠼⡟⠋⠉⠉⠛⠲⠀⠀⠀⠀⢀⣿⠇⠀⠀⠠⠤
⠈⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣶⣶⣤⣤⣤⣤⣴⣶⣶⣶⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠀⢀⣈⣉⣉⣉⣀
⠀⠀⠀⠉⠛⠷⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣴⣶⣶⣶⣦⡵⣶⣶⣾⠿⠛⠉⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣬⣭⣍⣉⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣉⣉⣉⣉⣩⣤⠴⠶⠛⠋⠉⣀⣀⣀⣀⣀⣀⠀⠀⣀⣈⣉⣉⣉⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⣶⣶⡿⠶⠷⠾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣷⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⣠⡴⠂⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⣿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠘⣿⣄⠀⠀⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠘⢿⣦⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣿⡆⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⡿⠁⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⢒⣚⣓⣋⣉⣉⣉⣉⣉⣉⣓⣚⡒⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣷⣶⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠓⢆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠊⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⣠⠤⠖⠒⢛⣉⣩⣭⣬⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣭⣭⣍⣉⡛⠒⠲⠤⣄
⠀⠀⠀⠈⠣⣄⣀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⠿⠟⠀⣀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠦⢬⣭⠭⣉⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣩⠭⢭⡤⠴⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠦⠤⠤⠤⠤⣤⣤⣥⣤⡤⠤⠤⠤⠤⠖⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣒⣒⡻⠭⠭⠭⠿⠿⠓⠒⠒⠦⠭⠭⠭⠭⣍⣛⣒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠋⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣭⣓⢤⡀
⠀⠀⠀⠀⠀⢠⡟⣇⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀⣀⠀⠀⠀⠉⠳⣣
⠀⠀⠀⠀⠀⠸⡟⢮⣑⡲⠤⢭⣍⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⣤⣤⣒⠾⡿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠉⠓⠒⠒⠪⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡏⣉⣉⣉⠓⢦⡄
⠀⠀⠀⠀⠀⠀⢸⣄⣀⡈⠙⠓⠶⠦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠖⠚⠉⣀⣼⠛⠉⠉⠉⠳⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡏⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⠁⠀⠀⠀⠀⣰⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢀⡴⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡴⠚⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠲⣄⠀⢀⣾⣠⣤⡴⠞⠋⢀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⢧⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢀⡌⠀⣬⠋⠁⢀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣠⣶⠶⠷⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠮⠭⣭⣍⣉⣛⣛⣋⣩⣭⠭⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢒⠊⠉⣉⣉⣀⣀⣀⣤⣤⣤⣤⣄⣈⠛⠛⠛⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⠛⠻⠿⠿⣿⣿⣛⣛⣛⣛⣛⣿⣿⡯⠷⠗⠚⠋⢹⡇⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⡀⢀⣆⠀⢸⡿⣋⡴⠚⠓⠲⣍⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢩⡀⢸⣇⣿⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡆⠀⣼⠉⠁⠀⠀⠀⢀⡼⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠀⠔⢠⡟⠀⢀⣀⡠⠴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⣾⡷⣚⣭⣵⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣾⢋⡵⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠒⠚⢳⡀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⣩⠅⣼⠋⠉⢛⣛⣷⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡅⠀⠀⠠⠴⠆⠘⣧⠀⠀⠀⢻⣍⣉⣀⡤⠥⠶⢚⣻⣿⢛⠟⠛⠳⠤⠀⠀⠀⣹⠄⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠶⠾⠛⠛⠉⠀⠀⠀⡀⢀⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢯⣖⣒⣒⣒⠶⠶⠶⠶⠶⠶⠶⠶⢚⣛⣛⣫⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠐⠒⠒⠒⠒⠒⠂



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

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣰⡤
⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠂⠀⠀⠀⠀⠀⠒⠖⠻⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡖⠒⠀⠀⠐⠒⠒⠶⢶⣷⣿⡴⠟⢻⣶⡄⠀⠀⠀⠀⣿⣤⣤⠀⠀⠀⠀⠠⠀⡤⣤⣤⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠂⠀⠀⠀⠀⠉⠈⠏⣿⡿⠀⠀⠀⢹⢹⠀⠀⠀⠀⣿⣿⢷⣦⣠⡶⠄⣴⣄⣷⣄⣰⣿⡟⠛⠿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠁⠉⠀⠀⠃⠹⡟⠁⣿⣿⡇⠀⠐⡙
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⣿⡏⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⡅⠀⢿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢀⣼⠟⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⡃⠀⣼⣿⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣿⣧⣮⠏⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⢠⡇⢀⣭⣿⣇⣠⣾⠇
⠀⠀⠀⠀⠀⠀⢻⣧⢠⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⠀⠀⣀⣾⡇⢀⣿⣿⣿⠝⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡷⠀⠀⠀⠀⠀⢸⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠀⠛⠁⠛⠛⠹⣷⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⡓⠐⠒⣀⣉⣙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣍⣉⡙⠛⠙⢙⣟⣟⣿⣽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣿⣿⣿⣿⣖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⣫⡾⠋⠀⢫⡝⠀⠙⢷⣝⣄⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣶⡾⠛⠋⠙⠛⢷⡤⡪⢟⡻⢕⢤⡾⠛⠋⠙⠛⢷⣶⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⣷⡔⡈⢀⢔⡟⣠⣴⣦⣄⢻⡦⡀⢁⢠⣾⡿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⡐⠩⣱⣲⢿⣾⢏⡥⣖⣲⢬⡹⣷⠿⣖⣎⠍⢐⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⢿⣄⢀⢔⣻⣷⣟⠽⠚⢺⡕⢪⡓⠓⢯⣻⣼⣿⣢⡀⣠⡿⢷⣕⢦
⠀⠀⠀⠀⠀⠀⣼⣾⠋⠀⡠⡫⠛⣩⠏⡵⣺⣀⠠⣢⣇⣸⣴⠀⣠⣗⢮⠹⣍⠛⢝⢄⠀⠙⣷⢧
⠀⠀⠀⠶⠶⢾⣿⣿⣏⠶⣿⡱⠰⣿⢸⢹⡱⣉⣉⣹⣿⣿⣏⣁⣉⢎⡏⡇⣿⠆⢎⣿⠶⣹⣿⣿⠷⠶⠶
⠀⠀⠀⠀⠀⠀⢻⢿⣄⠀⠑⣕⣤⣙⣆⡳⢽⠊⠐⠛⡏⢹⠻⠀⠉⡫⢞⣰⣋⣤⣪⠎⠀⣠⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠳⣝⢷⣾⠋⠈⠫⣽⡟⣯⣣⢤⢼⡣⢜⡧⡠⣔⣽⢻⣯⠝⠁⠙⣷⡾⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠅⣐⡹⢽⣶⣿⣎⡓⠿⠭⢚⣱⣿⣶⠯⢇⣂⠈⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣾⡿⢃⢁⠈⠪⣧⠙⠻⠟⠋⣼⠕⠁⡈⠘⢿⣷⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠿⢷⣤⣄⣀⣤⡾⠳⢕⣮⣵⡪⠞⢷⣤⣀⣠⣤⡾⠿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠙⢝⢷⣄⠀⣜⣣⠀⣠⡾⣻⠋⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣻⣿⣿⣿⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠄⠀⠶⠂⠒⠒⠒⠒⣲⣶⢤⣀⣀⣀⡐⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⠀⠀⠀⡀⠠⣌⣀⣀⣤⠀⠁⠈⠀⠛⢋⠛⢿⣷⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⣃⠀⠀⠀⠀⢠⠀⢩⡀⠈⠩⠳⣤⣀⣁⢼⡿⢃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣐⠀⠀⠀⠀⠀⡠⣦⡀⢴⠭⢽⠟⠉⠰⣏⣹⡷⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠤⠖⠀⠈⠙⠀⠋⡟⣿⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠶⠒⠋
⠀⠀⠀⠀⠀⢀⡔⠛⠛⠉⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠷⠦⣤⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠹⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⣿⠵⣺⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⣛⣷⠋⡷⢲⣦⣤⣀⡀⠀⠀⠀⢀⢀⣠⣶⣟⢒⣏⣹⣿⣧⠆⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠅⢠⠈⢉⡙⠛⠛⠿⠛⠿⠛⠛⡿⣏⣯⣮⠉⠁⠀⢀⢀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠋⠁⠀⡀⡀⠀⠀⠀⠀⠀⠈⠀⣤⠄⠰⠎⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠁⠀⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⢀⣀⣤⣤⡤⠴⠶⠶⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠤⣤⣤⣀⡀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣠⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠙⠳⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠉⠙⠛⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠖⠛⠋⠉⠁⠀⠀⠀⠈⣧⣼⣧⣤⡀
⠀⠀⠀⣠⠞⢛⣛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⠃⠀⠀⠻⣄
⠀⠀⢰⡏⢰⡏⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⣴⠖⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⡆⠀⠀⠀⠀⣿⠀⣠⠤⣶⡾⠓⠙⣦⡀
⠀⠀⠘⣧⠸⣧⣀⢀⣸⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠃⠀⢀⣠⡾⠋
⠀⠀⠀⠘⠷⣤⣉⣉⣩⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠈⢿⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠟⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⢦⣀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⢀⣴⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⠀⠀⣟⠓⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠞⠋⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠈⠓⠲⠤⠭⣝⣛⣒⣒⣲⡶⠶⠶⣶⣚⣛⣉⡡⠴⠚⠀⢀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⣤⣤⣤⣤⣀⣀⣀⣈⣉⣉⣉⣉⣉⣥⣤⣤⡤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠤⠤⠤⠤⢤⣄⣈⡉⠙⠲⠤⣄⡀
⠀⠀⠀⠀⣀⡤⠖⢒⡒⠒⠦⢤⣀⠀⠀⢠⠎⠈⢇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡉⠓⠦⣄⡀
⠀⠀⢀⡞⣡⠖⠉⠉⠉⠁⠀⠀⠈⠙⠲⢧⡄⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡉⠓⢤⣀
⠀⠀⡾⠰⠃⢠⠴⠚⠋⠉⠉⠙⠲⡄⠀⠀⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠓⢦⡀
⠀⢀⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠟⠐⠚⠁⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠉⠳⣄
⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⢄
⠀⠀⢧⠀⠘⣆⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠱⣄
⠀⠀⠘⣆⠀⠈⠳⣄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢧
⠀⠀⠀⠈⢧⡀⠀⠀⠉⠓⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡆
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠳⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⢀⢺⣤⢏⡴⠲⡄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣈⠉⠁⠈⠳⠚⠱⣅⡼⠁⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠶⠖⠚⠛⠉⠁⠀⠀⢀⣠⠤⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡏⠈⠳⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⠤⡤⠤⣄⡀⠈⠲⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⡇⠀⠀⠷⠊⠁⢀⠇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠉⠳⣄⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠹⡄⠀⠀⣀⠄⠀⢴⠟⠀⠴⢦⢠⡋⣳⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⡆⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠉⠀⠀⠀⢯⡤⠀⠳⠞⠀⠉⠁⠀⠀⠀⠀⠀⣀⣤⡎⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠉⠛⠛⠛⠋⠉⡡⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢤⣀⣀⣀⣀⣠⡤⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠀⠀⢿⣿⣄
⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠈⢻⣿⣷⣄
⠀⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤
⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠂⠀⠀⢀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣤⣤⣄⣀
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠧⣸⠟⢰⣭⠇⣎⠃⣴⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠛⠀⢘⡆⠂⡜⠐⠃⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠛⠻⠿⣿⣿⣿⣿⣷⣄
⠀⢸⣿⣧⠀⠀⣠⣤⣌⠁⢈⣤⣤⣈⠀⣠⣤⣤⡄⣠⣤⣤⡌⢠⣤⣤⣄⢀⣤⣤⣤⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⡄
⠀⢸⣿⣿⠀⢰⣿⠀⢸⡇⢸⡇⠀⣿⡆⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄
⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⣾⡇⠀⣿⡇⣿⣤⣄⠀⣿⣤⣠⠀⣿⣧⣤⠀⢸⣧⣤⡄⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷
⠀⠀⣿⣿⡇⢸⣿⠀⢠⡄⢸⡇⠀⣿⡇⣿⠀⠀⠀⣿⠁⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⢹⣿⡇⠈⢿⣤⣼⠇⠸⣧⣤⡿⠁⣿⠀⠀⠀⣿⠀⠀⠀⢿⣧⣤⣄⢸⣧⣤⣤⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢸⣿⣧⠀⠒⠀⠒⠒⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠖⠒⠒⠒⠒⠂⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠤⠒⠁⠑⢤⡀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠢⡄⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠓⠀⠀⠀⠚⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠇
⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠬⠒⠀⠶⢂⡀⠀⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣀⣤⣤⢀⣶⣶⣠⣤⣤⠙⢿⡏⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡠⠟⠿⠿⠿⣿⣿⡿⠿⠿⠛⠛⠉⠀⢡⠖⠂⠒⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣷⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢹⣿⣿⡶⠁⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣓⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡻⣿⣶⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡏⠀⠀⠀⠀⡴⣿⣿⣿⣿⢻⡟⣿⣷⣄⠀⠀⠐⠛⠲⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢰⡯⣿⡇⠀⣠⡞⠀⣀⡉⢿⣿⣿⣼⣷⣶⣦⡤⠤⠔⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⠁⠀⣟⣶⣦⣌⠉⠀⠀⠉⠛⠿⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠄⠰⠂⣀⡀⠲⠀⠀⠀⠀⠀⢹⣷⣿⣹⣷⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⢠⣴⣿⣿⣅⡰⠀⠆⠀⠰⠦⠤⢟⡿⣿⣿⣿⠿⣿⣛⣿⣿⢿⣿⣿⡿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣠⣾⠿⣷⣶⡏⠀⠀⠀⢰⣿⣵⠇⠈⠀⣠⣞⢿⡇⠉⠀⠉⠒⠢⠤⠄
⠀⠀⠀⠀⠀⣿⠷⠾⠿⠟⠃⠀⠀⢀⣤⣸⣿⣿⠀⠀⣼⢟⣽⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣿⣿⠿⠋⠀⣾⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠿⠛⠁⠀⢠⣼⠿⠟⠉

⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣠⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⡄⠀⠀⠀⢀⠀⠀⡀⣠⠂⠐⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠠⠞⠋⣣⠀⠀⠈⠀⢼⣆⢀⡙⢀⠄⠀⠀⠀⠀⢰⠊⠇⠐⠁⢤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠁⠀⠀⠤⠐⠋⠁⠀⠀⠀⠂⠀⠉⠃⠀⠈⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⢼⣤⢀⠣⠅⢦
⠀⠀⠀⢀⡤⠖⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣰⢀⡰⠛⡷⣀⠀⠀⠸⡆
⠀⠀⠀⢨⡔⢁⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢦⠠⣀⠰⠁
⠀⠀⠀⠸⢿⠕⠁⠀⠀⠀⠀⠀⢀⢔⡪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⠪⢓⠤⠄⢠⡊⠑⠒⠂⠀⠀⠀⠀⠀⠈⠀⢤⢴⠟⠫⠁⠀⠀⠀⠀⠀⠀⠠⣆⡤
⠀⠀⠀⣛⣀⠞⠤⣀⠐⠀⡀⠀⠈⡄⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⢃⡠⠲⡕⠄⠀⠀⢀
⠀⠀⠀⠀⠓⠚⡴⠂⠂⠆⠀⠀⠀⠐⢰⠈⠒⠢⠤⠄⡄⠀⠀⠀⠀⠤⠤⠒⠚⠉⣾⠏⠉⠀⠘⠠⠁⠀⠀⣤⠻⠀⠁
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡯⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢠⡖⠒⠛⠛⠒⠂⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢱
⠀⠀⠀⠀⠀⠀⠘⠁⢀⣠⠤⠤⡬⡾⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣆⢸
⠀⠀⠀⠀⠄⠀⠀⢀⠈⠀⡠⠊⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣦⠀⠀⠀⣯⢸
⠀⠀⠀⠘⠀⠈⢹⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⣀⣻⣦⢿⣠⡔⠂⠀⠀⠀⠀⠀⢀⡴⠊⠀⡈⠋⢢⡐⢄⠎⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠈⠙⣻⡿⢉⣻⣶⡤⠀⠀⠀⠰⣊⠉⠄⣆⣠⣆⣀⠲⠋⣡⠞
⠀⠀⠀⠀⠀⠐⠇⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣽⠛⢿⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣷⣬⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢹⣿⢗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠬⠤⠬⠥⠤⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠓⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣯⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢳⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⠞⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⢟⣡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣛⣛⣛⣛⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠛⠒⠒⠶⠶⠶⠶⠶⠤⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢖⣋⣙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃
⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠉⣈⣉⡷⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⢯⣀⣉⠉⠉⠉⠒⠲⠤⣄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠶⠤⠭⠭⠭⠯⠿⠷⠿⠿⠯⠭⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⢒⠂⣒⠤⢄⡀⢀⠀⠀⢀⡐⠂⢤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⠍⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠦⢝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣶⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠯⠁⠒⢤⣤⢄⢀⣀⡀⠀⢀⣀⡀⠀⢀⣀⣀⣀⡀⣄⣄⡶⠝⠂⡠⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠒⠀⠠⠀⠐⠔⠙⠀⠂⠒⠂⠈⠁⠀⠀⠀⣨⡶⠋⠉⠛⣦⡙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠘⠿⠮⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡂⠀⠒⠎⢠⡍⢛⣄⡠⠲⢁⣔⣶⣆⠀⠀⣐⡆⣴⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⣠⣴⡞⢹⡛⡿⢈⠗⠊⠁⠀⠀⠀⠀⠀⣀⠘⣻⣏⡛⢻⣿⣿⡄⣰⣿⠛⠁⠀⠀⠀⢀⡛⠃⣡⣔⣡⡤⣄⡀
⠀⠀⡠⢊⠕⠡⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠵⡾⢭⣿⣿⡿⢽⠁⢙⣠⠀⠀⠀⠀⢀⡮⠖⣉⣿⣿⡉⠻⢷⣿⣷⢄
⠀⢸⢃⡀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠉⣿⡿⠉⠳⠀⠙⠀⠿⠒⠶⠀⢠⣾⣴⣤⣿⣍⠃⠁⠀⡀⠘⠫⢿⣷⡀
⠀⠈⢇⢄⠀⠀⠈⠑⣿⣥⣲⣶⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⣿⣭⡿⣛⣤⣶⡿⠃⠀⠀⡸⣹⠁
⠀⠀⠀⠙⢷⣄⡀⠈⠁⠘⠋⠛⠉⠀⠀⠀⠀⠀⠀⢦⣶⣶⣬⣬⣥⣶⣾⣼⣠⣶⡆⠈⠉⠙⠋⠹⠛⡵⠋⠉⠀⠀⠀⢠⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⡀⠀⠀⠀⠀⠢⢀⡉⠉⠙⠙⠛⠻⠿⢿⠿⢛⣡⠇⠀⠀⠀⠀⢀⣈⣠⡤⠀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠛⠓⠒⠒⠲⠶⠿⠷⠶⠶⠶⠶⠶⠿⠾⠒⠒⠚⠛⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠢⠡⠀⡒⣒⣐⣀⣐⡒⠐⢐⣒⣀⣠⣴⠶⠶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐
⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣠⠴⠐⢁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⡠⠀⠤⠂⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠐⠒
⠀⠄⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡠⠀⠀⠀⠀⠀⠆
⠀⠠⠀⠀⠀⠀⠀⠀⠨⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠀⠀⠀⠀⠀⠂⠐⠊⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠴⠏⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠁⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠘⠄⠰⠅
⠀⠀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡤⠢⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄
⠀⠘⠀⠁⠁⠑⠀⠀⠀⠀⢀⠒⠀⠃⠴⠁⠒⢲⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀⠐⠂⢄
⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠐⠌⠀⠚⠈⠈⠀⠸⠀⠀⠀⠀⠁⠃⠀⠈⠀⠀⠀⠀⠀⠐⠀⠐⠂⠐⠀⠒⢈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠐⠠⠤⠥⠤⠔
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⢄⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⠈⢳⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣄⠹⣦⡀⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⢻⣿⣆⠸⣿⣷⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣼⣿⣿⠀⣿⣿⣿⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢟⣵⣿⡿⠃⣰⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢟⣫⣴⣿⠿⠋⣠⣴⣿⣿⣿⣿⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠟⣋⣥⣶⣿⣿⣿⣿⣿⠟⢁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⣋⣭⣴⣾⣿⣿⣿⣿⣿⡿⠟⠋⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⣉⣤⣶⣿⣿⣿⣿⣿⡿⠿⠛⢉⣀⣤⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⣠⣾⣿⣿⣿⣿⡿⠟⢋⣁⣤⣶⣾⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣼⣿⡿⣿⣿⡿⢋⣴⣾⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⠁⣿⡟⣰⣿⠟⣉⡠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠹⣿⡄⢿⠀⣿⠣⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢿⣌⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠐⠒⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠶⢄⠀⠀⠠⣶⡾⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠉⠉⣻⣿⣿⣶⢶⣦⠄⠀⢀⡤⠖⠋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⣀⣀⣠⡴⣿⠖⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠁⢀⡏⢀⠤⠒⠒⢦⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡠⠆⠀⠀⠀⢠⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⡀⢀⣀⠴⠋⣰⠏
⠀⠀⢀⣠⠤⠒⠂⠉⠉⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠁⣀⣴⣟⡁
⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⣠⣴⠟⠋⠀⠀⠙⢦
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⢠
⠘⠧⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣼⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠈⠙⠒⠦⠤⣀⣀⡈⠙⠛⠉⠙⠛⠛⠋⠉⠉⠐⠒⠒⠒⠤⠤⠤⠤⠔⠒⠛⢿⣿⣿⣿⣿⠿⠛⠁⠀⠀⣀⣀⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠲⠶⠶⠦⠤⠤⠤⢤⣤⠤⠤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠒⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠒⠚⠉⠁
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⢀⠀⣀⣤⣄⣤⠤⠤⠤⠤⢤⣤⣤⣀⡈⠀⠀⠀⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠋⠀⠀⠄⠀⠤⠀⠀⠀⠀⠐⠞⠛⢛⡛⣿⡿⣿⣿⣦⣄⡁⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⠃⠀⣀⠀⠐⠀⠘⠧⠤⠶⠶⣿⠀⢀⣄⣀⡀⠈⠁⣶⠈⢻⣿⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠸⢇⡀⠀⠀⠀⠀⢠⡦⠀⠤⠅⠀⠀⠒⢛⣶⣤⣀⣰⣾⡿⢃⣄⣿⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢬⡜⡒⠰⠀⠀⠀⠀⢘⣀⣶⣀⡀⣼⡡⠬⡿⠛⠁⢰⣿⢨⣿⡶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠂⠀⠀⠀⢀⣤⡴⠀⠘⠛⠷⠀⠞⣹⢿⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢋⣼⣏⣀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣰⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⣉⣉
⠀⠀⠀⠀⠀⠀⠀⡴⠛⠋⠉⠀⠀⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⣛⠛⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠰⠘⠀⠀⠀⠀⢀⡏⢿⡿⠾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⢠⢜⣿⣿⡵⢒⡟⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠒⢼⢷⣷⠟⣻⣴⢦⣄⣆⡀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣠⣶⢏⢝⡿⢙⣿⣻⣡⠄⠀⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠿⠿⠶⢞⡁⠘⠛⠻⠿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⢿⡳⡞⣝⠿⠟⠋⠉⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣋⣠⠞⢀⣴⡦⠀⠀⠀⠉⠀⠈⠁⠀⠀⠛⠻⠾⠛⠛⠃⠀⢀⣢⡴⢲⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣀⣄⣀⣀⣀⡀⢤⣀⣀⡠⢴⣶⠓⠀⠟⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⢀⠀⠀⢸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠄⠊⠁⢒⣀⠀⢀⡠⢂⠎⠀⠀⢁⠄⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠁⠀⡔⠀⠁⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡶⠂⠀⠒⠀⠀⠄⠀⠤⠀⠀⠉⠭⠁⠁⠀⡇⠀⠀⣠⠚⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⡟⠛⠉⠵⠶⠒⣀⠀⡀⠀⠠⠀⠤⠬⠭⢤⠤⠀⢀⡁⢒⠀⠁⠀⠉⢬⣖⣮⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠻⣿⣿⣿⣶⡴⣶⣶⣦⣌⣀⣀⣈⣉⣉⣉⣁⣀⣀⣉⣠⣾⣶⠶⢴⣶⣲⣭⡵⠊⠁⣀⡤⠶⠚⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠃⠀⠈⠉⠁⠉⠀⠀⠀⠈⠉⠛⠉⠙⠛⠛⢛⡛⢛⣟⣻⣿⣿⣿⣟⡃⠽⣂⠀⣟⡡⠀⢀⣤⠤⣤⣀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠀⠴⠶⢖⣛⣛⣯⠥⠄⢨⠅⢠⡏⣠⠞⠉⠀⠀⠀⠙⣧⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠂⢀⣘⠛⢫⡭⠿⠿⢚⣋⣠⠀⣊⢘⡟⠁⠀⠀⠀⠀⠀⠀⢸⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠄⠀⠶⠖⢚⣛⣩⡭⠆⢈⡅⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠀⠐⠒⢀⣈⣉⣩⣥⠤⠖⠀⡠⠄⢸⠁⠀⠀⠀⠀⠀⣠⠞⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡉⠁⣈⡉⢉⣭⡥⠴⠶⢒⣃⠠⠔⠀⢸⠀⠀⢀⡠⠴⠚⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⠴⠶⢖⣛⣛⣭⠅⠖⣂⠀⣼⡥⠉⠀⠀⣀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢐⣂⠀⣛⣋⣭⣭⠥⠶⠂⠨⠅⢠⡏⣠⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠄⠤⠴⠶⢒⣚⣋⠁⡴⢀⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠒⣂⣐⣾⣿⣿⠟⠠⠍⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢤⡤⠤⠴⠶⠶⠲⠛⣦⣌⠂⠀⠀⠀⠀⠠⢆⠠⣶⠌⢴⣶⣶⣶⣿⣿⡉⢓⣠⣾⠷⠶⣶⣶⣶⣦⣤⣤⣤⣀⣀⡀
⠀⠀⠀⣠⠶⠞⠛⠉⠉⠀⢘⡋⠀⠀⠀⠒⠒⠒⠛⠻⢯⡷⠲⠄⠀⠀⠀⠀⣂⣸⣤⣼⠿⠿⠿⠿⠖⠾⢿⡿⢟⠛⠛⠛⠿⠶⠶⢦⣤⣉⠉⠉⠛⠻⣷⣶⣄
⠀⠀⠘⣧⡠⣄⣶⡀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠳⠄⠀⡀⠑⠒⠤⠤⢤⣀⣀⣀⣀⣒⣀⣾⣿⣿⣯⡼⡖⠋⠀⠤⠴⠂⠀⠠⠔⠀⢀⣀⠬⠃⢀⣰⣄⡠⢗⣼⠃
⠀⠀⠀⠈⠙⠲⣭⣝⣛⠒⠶⠶⠤⠤⢤⣭⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣠⣥⣤⣀⣀⣀⣀⣲⣶⣶⣶⣯⣥⣤⠤⠶⠶⠖⠚⣋⣩⣵⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⣤⣤⣤⣄⣀⣀⣀⣉⡉⠉⠉⠉⠉⠉⠭⠉⠉⠩⠭⠍⠉⠉⠍⠉⠉⣉⣉⣉⣀⣀⣀⣤⣤⣤⡶⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣀⡤⠚⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⢡⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢾⡁⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠈⢻⡆⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⣠⠞⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡚⠉⠉⠀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⠀⠀⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠱⣏⣤⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣭⣭⣿⣲⣦⣍⣦⢸⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⢸⠦⢤⡀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⡟⠀⠀⡷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⣿⠃⢀⡼⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⣰⡧⠖⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⢧⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⢠⠞⠁⠀⢠⠞⢡⡞⣽⣧⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⢠⣿⣾⡭⠴⠖⣒⣿⣧⣤⡤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠙⠲⣤⣙⠿⡴⠿⣍⣉⣉⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠾⠭⠭⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠢⢄⠀⠀⠘⣦⠀⠀⣰⠋⢀⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣐⡦⠀⢳⠀⠀⡾⠀⠀⡇⠀⠻⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡾⠀⠠⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠤⠤⠀⠲⠶⠶⠶⠶⠦⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⡂⣉⢡⢤⢄⢆⣄⣴⣄⣞⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣿⠉⠙⠓⠷⢏⣉⣉⣉⠉⠉⠀⠀⠈⠈⠀⠀⠀⠉⢉⣉⡉⠙⢒⣒⣤⠤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣿⠀⠒⠦⠤⠄⢀⣈⣉⠉⠉⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⢻⡏⠉⠁⠐⢚⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⠋⠉⢹⡆⢉⣉⠒⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣩⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⡀⠀⠈⣧⠀⠬⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠂⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢧⡀⠀⢻⡀⠢⠤⠀⢉⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡌⠳⠦⠼⣧⠀⠒⠀⠀⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠳⠖⠚⠳⣄⠁⠄⠐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣳⣦⣘⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⢦⣀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⢒⡊⠉⠉⠀⠀⠀⠀⢸⢟⠋⠒⠒⠶⠤⠤⠤⠤⠤⠤⠶⠖⠒⠉⠁⢸⠉⠉⠉⠉⠉⠑⠒⢒⠠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠵⠒⠉⣉⡩⠥⠴⠒⣀⠀⠀⠀⠀⠻⠦⣷⣃⣢⣐⣂⡠⠀⢂⣀⣀⣀⣀⡤⠴⠊⠀⠀⠀⠀⠀⠐⣀⣨⠍⢓⡄⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠥⣤⣄⣀⣉⣁⠐⠛⠀⠴⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⣐⣒⣫⡭⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⡠⡤⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⠋⠀⢀⡆⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⢀⠀⠉⠢⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⣴⣵⣿⡿⣿⣤⡀
⠀⢰⣶⡄⠠⠸⠦⢠⠀⣶⡆⠀⠀⠀⠀⣤⠤⡤⢤⡤⣬⠥⣤⠤⣤⢤⣄⠀⠀⣿⣿⡼⢮⠤⠾⠼⣼⢤⣿⣿⠀⠀⠀⠀⢿⣸⣯⣅⣦⣩⣩⡗
⠀⠈⢿⡷⣄⡀⢀⣊⣼⣿⠃⠀⠀⠀⠀⠻⠶⠓⠞⠷⠾⠰⠗⠞⠗⠾⠗⠀⠀⠙⠿⠷⠶⠶⠶⠶⠶⠾⠿⠟⠀⠀⠀⠀⠀⠑⠿⠿⠿⠗⠁
⠀⠀⠀⠙⠻⠯⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡔⣁⣠⣶⣴⣄⣁⢂⠀⠀⠀⠀⠀⢀⡴⠛⢉⣭⡉⠛⢦⡀⠀⠀⠀⠀⠀⠄⣨⣴⣶⣦⣬⡀
⣶⣶⡶⢦⢶⡶⠶⠴⠶⠶⢶⣦⡆⠀⠀⣠⣾⣏⣉⣍⣉⣩⣍⣩⣹⣧⡀⠀⠀⣠⣿⣄⣀⣠⣤⣄⣀⣡⣿⣄⠀⠀⢀⠶⡚⣿⣽⠋⡏⡸⡟⠾⣒
⣿⣿⣮⣷⣥⣤⣧⣼⣤⣥⣼⣿⡇⠀⠀⠛⡁⢺⠲⠷⠇⠟⠻⠽⠅⣸⠃⠀⠀⢿⡧⣶⣤⣤⣧⣾⣼⡤⢿⡿⠀⠀⢸⠸⢿⡁⣤⢸⣇⡧⣼⡷⠟
⠈⠉⠀⢀⡀⠀⠀⠀⢂⠐⠀⠉⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⣿⣿⠃⠀⠀⠀⠀⢳⣜⠿⣯⠿⣿⠿⣣⠟⠀⠀⠀⠈⠙⡙⢿⣿⣟⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢀⣀⡠⠒⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⡀
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⢀⠅⠀⠚⠒⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⢀⡤⠒⠋⡘⠒⢄
⠀⢠⠂⣔⢁⡤⡶⣖⡠⣠⠈⢢⠀⠀⠀⠀⡇⡠⠤⣀⣠⣴⢀⠀⡁⢄⠀⠀⣀⠀⠸⢭⠤⠄⢹⢹⠤⢀⠄⢛⡛⠀⠀⠀⡈⠀⠀⢸⡁⠀⠈⡃
⠀⠘⢄⢈⡤⠗⢓⠃⠁⠁⢁⠊⠀⠀⠀⠀⡆⠳⠐⠋⠟⠟⢛⣙⠁⠔⠀⠀⠉⠐⠤⠒⠼⢇⣏⡸⠼⠧⠟⠈⠁⠀⠀⠀⢠⠀⠀⠠⠀⠃⢀⠁
⠀⠀⠸⣄⠂⠀⠛⠃⢀⢂⠄⠀⠀⠀⠀⠀⠀⠆⠢⢀⠛⠃⡠⢄⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠑⠠⠌⠡⠄
⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⡠⢰⣐⣀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠠⠋⢸⣿⣿⣿⣿⠈⢂⠀⠀⠀⠀⠀⢀⠐⠈⢁⣄⡤⠈⠀⣀⢰⡀⠀⠀⠀⡀⠀⣀⣴⣶⣤⡀⠀⡀⠀⠀⠀⠀⠀⣀⣂⣆⣹⣯⣤⣶⣶⣶
⠀⣾⣟⢻⠛⡟⣿⢻⠺⡟⣿⣷⠀⠀⠀⣰⠉⢡⠶⢾⢹⠳⡟⠻⠍⠟⠁⠀⠀⣿⣿⣏⣯⣿⣿⣏⣏⣿⣩⣿⠀⠀⣶⣿⣿⣩⣿⣿⣽⣿⡿⠿⣿
⠀⠈⢍⠉⢹⣿⣿⣯⣿⠉⠩⠉⠀⠀⠀⢘⣃⣨⠥⢿⣚⠷⠻⣽⣿⠁⠀⠀⠀⠈⠉⠉⠙⠻⠿⠟⠉⠉⠉⠁⠀⠀⡾⠟⠛⠉⠉⠥
⠀⠀⠀⠃⠜⢛⠿⢟⠋⠔⠁⠀⠀⠀⠀⠉⠉⠀⠋⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣉⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣋⣩⣥⠤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠤⢤⣍⣙⡓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠿⠖⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠓⠶⣌⡙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⡿⠋⠁⠀⠀⣀⡤⠴⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠶⢤⣀⠀⠀⠙⣦⠹⡆
⠀⠀⠀⠀⠀⠀⣾⢸⡇⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⠳⣄⠀⠀⠀⠀⠀⠈⠙⠦⣄⡿⢀⡇
⠀⠀⠀⠀⠀⠀⡿⣎⠳⣏⡁⠀⠀⠀⠀⠀⠀⢠⠶⠚⠋⠉⠉⠉⠀⢠⠖⠲⣄⠸⡆⠀⠀⠀⠀⠀⢀⣠⢟⣡⠟⣇⣠⠤⠶⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⢿⠈⠳⢬⡙⠶⣤⣀⡀⠀⠀⢸⡄⠀⠀⠠⡞⠛⠛⠋⠀⠀⣹⢠⡇⠀⣀⣠⡤⠖⣋⡴⠋⣡⠞⠉⠀⠀⣀⣀⣀⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠓⠶⢬⣉⡛⠒⠶⠿⢦⣤⣤⣽⣦⣤⣤⣤⡤⠿⠾⠛⠋⣉⣠⠴⠚⠁⠀⠀⣯⣀⣠⠴⠚⠉⠀⠀⠈⠙⢦⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠙⠓⠶⠶⠶⠶⠒⠋⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⣄⣀⣀⣀⣠⡤⠴⠚⠉
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⣄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣟⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⠁⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⠙⠲⠦⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠚⢉⣡⠼⠃⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠳⣝⡲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⢭⣛⡒⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⢖⣚⡭⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠾⠭⠭⠭⠭⠭⠭⠵⠶⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣄⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣒⣛⣉⣉⣉⣁⣀⣈⣇⣀⣀⣀⣀⣀⣈⣉⣉⡓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⠁⠀⠀⢴⣒⣒⣛⣛⣉⣀⣀⣀⣀⣀⣀⣤⠤⠞⠛⣲⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠓⠤⠤⣤⣤⣤⣤⣤⣤⠥⠥⠤⠴⠒⠒⠒⠛⠉⠉⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠉⢙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⡤⠖⠋⢰⠇⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡜⠁⠀⠀⠀⣼⢀⣴⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢦⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠙⠲⢤⣄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣯⣛⣿⣿⣿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡴⣒⡒⢦
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠄⠈⠂⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣄⠀⢳⠖⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⢈⡳⣄⠱⡄⠀⠀⠀⠀⠒⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣼⡤⠔⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠙⠓⠲⠾⠧⠬⠦⠤⠤⠤⠤⠾⠥⠬⠿⠞⠛⡖⠒⠚⠋⠉⠀⣼⠔⠚⠛⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢀⡰⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⢁⡴⢋⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⣯⠞⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⣸⣷⣒⣉⣀⣀
⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⡧⠤⠄⠀⣈⣉⣉⠉⠓⠲⣄
⠀⠀⢠⡴⠋⠁⡀⠤⠐⠒⠚⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣼
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠦⠤⠤⠼⡤⠴⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⢒⡿⠁
⠀⠀⠘⢯⡒⠲⠦⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣱⠤⣄⣤⡤⠤⠴⠖⠒⠚⠋⢉⣉⣁⡤⠴⠒⠋
⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡾⠀⠀⠀⠑⢦⠶⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢦⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢀⣾⣿⣿⠟⠉⠉⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿
⣾⣿⣿⣿⡀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠟⠛⠉
⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣍⣀
⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣦⣄⣀
⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠙⣿⣿⠉⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣿⣿⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣌⣥⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⢾⣿⣿⣿⢿⡷⠶⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣕⣺⡿⠿⠿⢯⣿⣿⠾⠯⠿⣿⣒⣭⢀⣀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡶⣹⠇⠀⣠⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡲⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣱⣶⣚⡭⣠⢞⣵⡿⠁
⠀⠀⠀⠀⠀⠀⣴⣿⢿⡻⣿⣯⡽⣨⡯⣽⢿⣽⣿⣿⣭⠽⣸⣯⣻⡿⣫⡾⣳⣿⢏⣄
⠀⠀⠀⠀⠀⠀⠻⣿⣮⣵⣿⣿⣾⣿⣿⡿⢿⣛⣓⣛⣛⣯⣭⡶⣫⢾⢟⣔⣞⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠯⣝⣛⡿⠿⢿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⠮⠵⣟⣛⣭⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠒⠒⠒⠒⠀⠀⠠⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠀⠀⠀⠀⠒⠂⠀⠈⠉⠉⠁⠀⠒⠂⠀⠀⠠⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠠⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⠌⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣆⠀⢁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⡩⡆⠀⡭⠘⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣅⠂⢄
⠀⠀⠀⠀⠀⠀⡔⠁⠐⠁⠂⠙⢄⠑⠤⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡠⠄⠀⠀⢸⣕⢄⠑⡄
⠀⠀⠀⠀⠀⢰⢠⠁⠀⠀⠀⠀⠀⡱⠢⠄⢉⠉⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠕⠁⠀⠀⠀⢀⠌⠀⠁⢡⢰
⠀⠀⠀⠀⠀⠈⣌⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠙⠖⠃⠀⠠⠤⠤⠤⠀⠀⠀⣀⠡⢆⡰⠦⢤⣤⠤⠖⠉⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠐⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢒⠲⠶⠒⣈⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠉⠐⠀⠠⠀⠀⣀⡀⠀⠀⠈⠀⠀⠐⠈⠀⣀⣀⣠⣤⡤⠤⠶⠒⠚⠉⠀⡢⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢟⣫⠭⠖⠒⠒⣒⡒⠲⠯⣝⡻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⠖⠋⠀⠀⠀⢻⣶⢈⣿⢠⠀⠀⠉⠳⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⠞⠁⠀⠀⠀⣠⣤⡼⡫⣚⣵⢟⡤⠀⠀⠀⠈⠳⡝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢣⠏⠀⠀⠀⠀⠐⣿⣹⢇⣿⡽⣺⠋⠀⠀⠀⠀⠀⠀⠹⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⡏⡞⠀⠀⠀⠀⠀⢀⣀⣉⣙⣏⣛⣵⣶⣂⠀⠀⠀⠀⠀⠀⢹⢹⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠿⡇⠀⠀⠀⠀⣷⠙⠿⠿⣿⣻⣻⣿⣿⣯⣵⣆⣀⣀⡀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⡀⣿⣾⡇⠀⠀⠀⠀⢿⡗⠀⠀⠀⠈⢹⣿⣿⣿⣿⡿⠋⢙⣿⠀⢸⣷⣿⢀
⠀⠀⠀⠀⠀⢰⣭⣻⣿⣇⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣴⠿⠋⠀⣸⣿⣟⣭⡆
⠀⠀⠀⠀⠀⠀⠹⣷⣿⣻⣷⡄⠀⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⠀⢠⣾⢿⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣶⣬⣝⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⣫⠥⢶⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣷⣶⣽⣿⣶⡒⣲⣖⣒⣶⣿⣿⣶⣾⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠙⠻⠾⣿⣿⣿⣿⣿⣿⡷⠟⠋⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠛⠳⠿⠭⠭⠭⢽⣭⣤⡤⠤⠤⠤⠤⠤⠖⠚⢛⡆⢀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠖⠒⠒⢄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠷⠁⠀⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣳⣶⣿⠀⠀⠀⠀⠀⣸⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣽⢷⣿⡇⠀⠀⠀⠀⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠂⡀⠈⠀⠈⠀⠀⠀⠀⠰⠀⢀⠻⢿⣛⣻⠁⠀⢀⡠⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⢟⣒⣭⠵⠞⠉
⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡾⠋⠀⠀⠀⠀⠀⠀⠀⣈⣴
⠀⠀⠈⠙⠒⠶⢶⣤⣤⣤⣤⣌⣧⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⣀⣼⣤⣤⣤⣤⣶⣶⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢿⣿⣿⣿⡯⢉⣉⣉⣉⣙⣛⣛⣿⣿⣭⣭⣽⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣉⡩⠶⠶⠾⠿⠿⠶⠶⠶⢍⣉⣵⣿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⢛⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢀⡼⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⢀⠞⠁⠀⣼⣿⣿⣿⣿⡇⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣌⣀⣠⣾⣿⣿⣿⣿⣿⡇⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉





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

⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⢀⢸⣦⣴⣶⣿⣿⣿⣿⣿⣭⣉⣿⣿⣿⣿⣿⣶⣬⣽⣍⣭⣿⣫⣟⡻⠿⢿⣶⣤⡀⢰⠄⡄⠀⢀⡀⠤⠀⡀
⠀⠀⠀⢸⣦⠙⠿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠿⢋⣼⣧⣞⣋⠀⠀⠀⢼⡆
⠀⠀⠀⠈⠛⠈⠉⠈⠙⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢛⣩⣵⣾⢿⣿⣿⣿⠁⠀⠀⠀⠈⡏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⣿⣿⡿⢸⣿⠿⠃⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⣾⠃⠀⠀⠀⢀⡀⢁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⠃⠀⣀⠠⠔⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⢋⣡⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⡾⠛⠋⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠩⠝⣓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠠⣷⡆⠈⠐⠠⢄⣀⢀⣤⠤⠖⠒⠒⠂⠘⠒⣂⣀⣀⣀⣀⣀⡀⠒⠒⠲⠤⢤⣄⣤⣤⠶⠛⠉⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢮⡇⣤⡴⠓⠀⠀⠉⠙⠛⠒⠶⠒⠒⠠⠀⣨⠽⠟⠛⠛⢛⣛⠛⠛⢚⡛⠛⠣⠀⢈⠑⠲⠀⡇
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠈⢳⡿⣿⡄⠀⠀⠀⠄⢀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⠤⡙⠛⠀⠀⠀⠀⣀⡤⠁⡀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠈⠒⠭⢳⡶⣦⣤⣄⣀⣀⣈⣁⣀⣉⣉⣉⣉⣉⣈⣀⣀⣵⣶⣶⠶⠾⠛⣋⣀⢤⡏⣹
⠀⠀⠀⠀⠀⠀⠀⠐⠙⢿⣷⡄⠀⠀⢀⢳⠀⠀⠈⠉⠈⠙⠉⠭⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⡀⠀⠈⠈⠹⠇⠛⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣴⣶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄⠀⠀⠀⠀⠠⠀⢀⢢⣤⣄⢀⠀⠀⠀⠀⠀⠀⢠⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢄⠀⠀⡆⠰⢾⡺⢀⡝⣪⡝⠶⠦⢤⡀⠀⣦⡎⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠶⠛⢿⢻⣿⣦⡀⠀⠆⣀⠼⠰⠈⢷⡝⢇⠘⠀⣴⡄⢠⡟⡇⠀⠀⣠⢟⣵⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠐⢦⠀⠙⠽⣿⣿⣦⣄⠈⠀⠐⠰⠦⣤⣄⣀⠘⠋⠃⠸⠁⢁⣤⣾⣵⡟⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠁⠖⢚⣩⣡⠤⣟⠻⣶⣤⣄⣀⣀⣀⣀⣀⣤⣶⣶⣾⣿⣿⡏⡿⠾⣶⣤⣄⡀⠀⠀⠀⠀⣀⣴⣶⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠻⠶⣤⣀⣠⣶⣿⣿⣿⣿⣿⣿⣕⡮⣍⣙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣠⣷⣤⣤⣿⣿⣿⣟⣷⣶⣿⡿⠟⢉⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣈⠉⠉⠛⠛⠻⠶⠶⠤⠿⠿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠽⠶⠾⠿⠿⠛⠛⠛⠉⣉⣁⡤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠶⢶⡶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠛⠛⠋⠉⠉⠛⠛⠓⠒⠲⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣤⠀⠀⠀⢀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⢦⣄⣠⠾⠻⣤⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⢀⣾
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠋⠁⠀⠀⠀⠀⠀⣠⠟⢸
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⣤⣤⣤⣤⣀⣻⣄⠀⠻⣦⡀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡿
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣛⣳⣶⣦⣤⣤⡈⠉⠁⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⣸⠇
⢸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠁⠀⠀⠀⠀⢠⡏
⢸⡀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⣛⣋⣉⣉⣁⡤⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤⠤⠔⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠘⡇⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢩⠿⢛⣉⣥⠴⠖⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⢹⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣡⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⢳⡄⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠦⣤⣤⡴⠞⠋⠀⠐⠟⠒⠛⠛⢛⣋⣩⣭⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠓⠒⢲⡆⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⢯⣭⣉⡉⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠲⠶⠶⠦⣤⣄⣈⡛⠳⢶⣶⡶⠦⠬⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣲⣤⣽⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⡤⣄⠀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⣲⡤⣵⡦⠂⠙⠓⡉⠉⡉⣉⣙⠛⠲⠶⢾⣻⣯⣕⣢⡤⢄⡀⠀⡴⣣⠬⢦⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⣟⣍⢱⠤⠀⠉⠉⠀⣀⠀⣤⠢⢆⡹⢿⢉⡤⢤⡀⠀⠉⠉⠁⠈⡉⠻⣿⣶⣸⡇⠀⠀⠈⠈⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⢟⣶⣄⠀⠀⢠⣤⣿⣿⣼⢿⣯⠤⡝⠼⡓⠀⠈⠁⠀⠀⣀⡴⣩⢮⣽⡷⠁⣿⡀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠑⡘⣛⣥⡁⠀⠈⠸⠞⠋⢋⡀⣯⡀⠛⠅⠉⠀⠀⠀⠀⠀⠈⡋⢁⡬⠾⣜⡷⠷⠆⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠳⢶⣭⣵⣅⡀⢀⣀⣀⣀⠀⣁⡈⣂⣀⣀⣀⣠⣐⣦⣄⠼⠷⠛⠋⠀⠒⠆⢃⠘⠂⣀⠤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢉⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠙⠉⠋⠛⠙⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠨⠓⢈⣤⡴⠊⠑⢄⣀
⠀⠀⠀⢰⣿⠁⠀⡖⢸⡴⢖⡤⠁⠀⠀⠀⡴⠀⠰⠇⡤⠀⢀⣠⡴⣄⡀⠀⠀⢠⣻⡿⠂⠀⠀⠀⠀⡡⠪⡔⢲⠦⠄⠀⠘⠷⣶⡄
⠀⠀⢀⣿⠃⠀⠀⢄⣠⡶⣦⠌⠀⠀⠀⠼⠿⡶⠔⠛⣁⡛⠉⡥⣿⣬⠑⡴⢢⣤⠴⠵⠂⠀⠀⢠⡴⠀⠒⠓⡉⠀⢠⢠⣴⣠⡟⡇
⠀⠀⠀⢿⣿⡶⠄⠈⠻⠝⠈⠋⣠⠄⠂⠀⠀⠠⠈⠉⠁⢽⡏⠃⠁⠙⣠⠶⢿⠀⠀⠀⠀⠀⠀⢲⣛⡠⠀⠀⠀⠀⢈⣭⣾⡽⠚
⠀⠀⠀⠀⠙⢶⣆⡀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⠀⣄⡠⠀⠀⣀⣠⡤⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠓⠶⠴⠤⠤⣄⣀⡀⠀⠸⢅⠒⠠⠤⠤⠴⠦⠤⠦⠵⠶⠒⢋⣜⣠⡤⠬⠤⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠉⠉⠉⠉⠉⠉⠛⠒⠛⠋⠉⠉⠉⠉⢉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠛⠛⠛⠛⠻⠛⠉⠙⠋⠉⠉

⠀⠀⣴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠙⡢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠄⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⢤
⠀⢹⣿⣿⣷⣛⠇⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⡾⢧⠤⠃⠀⠀⠀⠀⠀⢸⣿⣻⣧⣾⢟⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⢻⣿⢀
⢴⣖⣹⣿⣱⣷⠄⠀⠀⠀⠀⠀⠸⣿⣇⣿⣿⣨⣿⠗⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠿⠁
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⡂
⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⢠⣶⣿⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣳⠄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣧⣤
⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣽
⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠆
⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇
⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⣀⣀⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⠾⠿⠛⠛⠛⠛⠛⢻⡟⠛⠛⢻⣿⠛⠛⠛⠛⠛⠻⠿⠷⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣀⣀⣀⣽⡇⠀⠀⠀⠈⠉⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠋⠁⢸⣷⣤⣤⣄⣀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠻⣷⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡼⠃⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠶⠶⠶⠛⠉⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣤⣶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠾⠿⠛⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⠛⠳⠶⢦⣤⣀
⠀⠀⠀⢠⣾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⣻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠙⠻⢶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠚⠉⠀⢀⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢤⣤⣄⣉⣉⣉⣛⣛⣛⣉⣉⣁⣤⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣽⣛⣛⣛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⣛⣛⣛⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠷⠶⠶⠶⠶⠾⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⢟⠿⠋⠁⠤⠀⠠⡈⠉⠁⠒⠦⢍⡚⢕⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⣿⡕⠁⡢⠔⠈⠀⠀⡨⣃⠀⠀⠀⠀⠐⢌⠓⣝⢷⣬⠷⠐⠠⣄
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⣾⠏⡄⡜⠁⠠⠚⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠑⠈⠆⢿⣦⠴⠒⢘⠂
⠀⠀⠀⠀⠀⢀⡞⡔⠁⠀⢸⡸⠘⠈⠀⠀⠈⠉⠀⠀⠀⢤⠀⡈⢁⢠⠈⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⢷⢠⢀⠀⡀⡀⠀⠈⠢⠠⢀⡌⠀⢀⠀⡀⠸⠀⠀⠀⠀⣇⣿⣥⣀⡀⠀⠀⠉⠐⠆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢻⡸⡌⠆⢆⢰⣔⢄⡀⠀⠀⢀⠁⡴⣡⢘⠔⠁⠀⠀⡜⡸⣹⣿⡋⠙⠍⠟⡖⠦⠤⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣧⡱⡄⠀⠀⠙⠻⠏⣛⠲⣦⡁⣴⠿⠃⠀⠀⢀⠜⡰⢡⡿⠉⠁⠀⠀⠐⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⣜⢦⡈⠓⠤⣀⠀⠉⠂⢑⠃⠀⠀⠀⢐⡡⢞⣴⡿⠁⠀⠀⡀⠀⠀⣸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢮⣓⠦⢤⣉⣉⣀⣀⣀⠤⠴⣒⣩⠖⣫⠟⠁⠀⠀⠀⠁⠀⠀⡿
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠁⠹⡷⣬⣍⡓⠒⠒⠲⠒⠒⠚⢋⣡⠴⠛⠁⠐⠐⠀⢀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠡⡡⡈⠢⣙⠻⠛⠓⠚⠚⠛⠛⠉⠁⠀⠀⠀⠊⡠⢠⢊⠄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠊⡂⠀⠑⠀⠄⡀⠀⠀⠀⠀⢀⠠⢐⠠⢊⠔⠱⠃⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠊⠐⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢽⣶⣦⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⢴⣒⡭⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠯⠽⠿⠿⠿⠖⠒⠋⠉⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠒⠤⢄⡀
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⢠⠇⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⢇⠀⢀⠇⢀⠴⣶⣄⠀⢀⠴⢶⣄⠀⢇⠀⢀⡏⠀⠠⠤⡀⠀⢀⣠⣄⠀⠀⢧
⠀⠀⢸⠀⢰⣉⡿⢷⠀⢰⣉⣹⢷⠀⢸⠀⢸⠀⠸⣶⣿⡽⠀⠸⣶⣾⣹⠀⢸⠀⢸⡀⢰⠉⣻⣮⠀⢸⣉⣹⢷⡀⢸
⠀⠀⠈⢇⠘⠻⠿⠋⠀⠈⠻⠿⠚⢀⠇⠀⠀⢇⠀⠈⠉⢀⠤⠤⠈⠉⠀⢀⠏⠀⠀⢇⠘⠿⠷⠞⣠⣌⠻⠿⠚⢀⠎
⠀⠀⠀⠈⠣⣄⠀⠂⠀⠐⠁⢀⡴⠋⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠳⣄⠀⠈⠁⠀⠁⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡞⢀⠤⣥⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡜⢀⣠⣄⡀⠀⠀⢀⣤⣀⠈⡆
⠀⠀⢸⠀⠠⡀⠀⣀⠀⡀⠀⣀⠀⠀⢹⠀⢸⠀⢺⣦⣟⡻⠀⠤⣀⡠⠆⠀⢸⠀⢸⠁⢾⣤⡟⣷⠀⠐⣧⣼⣛⡇⢸
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡞⠀⠘⡆⠀⠉⠉⠀⣀⣀⡀⠀⠀⠀⡜⠀⠈⡆⠈⠛⠛⢁⡀⢀⡉⠛⠋⠀⡜
⠀⠀⠀⠙⢄⠀⠀⠒⠦⠒⠉⠀⢀⠜⠀⠀⠀⠘⢄⠀⠀⠉⠀⠀⠁⠀⢀⠜⠁⠀⠀⠘⢄⠀⠀⠻⣿⡿⠃⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠴⠒⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠒⠒⠒⠒⠒⢒⢖⡶⡦⠄⠀⠠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠯⠳⠉⠓⠀⠀⠀⠀⢰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⢠⡀⠀⠀⠀⢀⡀⣀⡈⣐⡀⠈⠀⠀⣀⠠⢠⣬⣶⣗⣇⣠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⢿⣷⠺⣷⣖⣃⣠⡞⣿⣿⣳⡥⠄⠀⢤⣔⢳⡖⢺⢿⡟⣸⣷⡵⠟⢛⣿⡇
⠀⠀⠀⣀⡀⢤⡤⢶⠂⣋⢺⠙⠀⠙⢶⣥⣼⠅⠉⠛⠉⢹⢛⣃⠳⠽⡌⢫⡩⠄⢃⣿⠁⠀⣀⣼⡿⠁
⠀⣰⣿⣷⣦⡀⣝⢢⡁⠼⣷⡀⠀⠀⠀⠙⠫⠉⠉⠀⠀⠀⣠⡇⠁⠀⠀⠈⠉⠀⣍⡮⣊⣍⣿⡟⠱⠒⠤⣀
⠈⢿⣿⣿⣿⣽⡛⠃⠈⠂⠘⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠓⠀⠀⢀⠀⣀⣴⡿⠞⢛⣿⣿⡵⣂⣴⣶⣜⡆
⠀⠈⠻⢽⣻⡾⣿⣿⣷⣶⣶⣤⣿⣿⣶⣷⣸⣤⠀⣰⣆⣄⢠⣤⣴⣣⣾⠶⠋⠁⠀⠀⠠⠖⣛⣓⠿⢟⡿⠋
⠀⠀⠀⠀⠉⠻⠿⢶⣯⣭⣭⣛⣛⣿⣿⣿⣯⠿⠯⠝⠛⡉⣉⢉⣉⣛⣁⣤⣐⣒⡶⠭⣉⣴⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣯⣅⣠⣶⡶⠤⠼⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀
⠀⢠⣴⣾⡿⠿⢟⣛⣛⣫⣭⡭⠭⠭⠭⠭⠷⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⣛⡛⠻⠿⢿⣷⣶⡄
⠀⣿⣿⢷⣾⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣷⡾⣿⣿
⠀⣿⣿⠀⠀⠉⠭⠟⠛⠛⣛⣲⣶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠶⠶⠶⠶⠶⠶⢶⣖⣛⣛⣛⠿⠉⠙⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣶⣾⣿⠿⠿⠿⣿⣷⣦⣄
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠋⠉⣀⣤⣤⣤⣤⣤⣈⠙⢿⣷⣄
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣾⡿⠛⠋⠉⠉⠛⢿⣷⡄⢹⣿⡆
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⣿⡀
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⠃
⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣾⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣰⣿⠇
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣰⣿⠏
⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⣼⣿⠋
⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢀⣠⣴⣿⠟⢁⣴⣿⠟⠁
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⣴⣾⡿⠟⠋⣁⣴⣿⠟⠁
⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠉⠁⢀⣤⣾⡿⠛⠁
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⣤⣾⡿⠟⠉
⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉
⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠉⠁⠈⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⣴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠹⡜⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢹⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⠃⣾⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠊⣸⠇⠀⢸⢹⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢠⡏⠀⢠⠇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣾⠁⠀⠋⢠⡟⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⢿⡀⠀⠤⣾⠁⠘⠁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠟⣛⣉⣉⣤⡤⠌⠳⣄⠠⠿⣄⠠⢤⣤⣤⣄⣈⠉⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣋⡉⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⢉⣃⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⠉⠙⠛⠛⠻⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠉⠉⢀⣸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⡇⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡥⠞⣩⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠶⠶⠿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣤⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀⡀
⠀⢰⡟⠉⠀⠀⠀⢀⣠⣴⣶⠾⠿⠛⠛⠋⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⢣
⠀⠸⣷⡀⠀⠀⢰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⠀⡼
⠀⠀⠙⢷⣦⣀⠈⠻⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠃⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⠛⠿⢶⣤⣬⣉⣛⠛⠻⠿⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠤⠶⠴⠶⠒⠛⠋⠉⠁⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⠶⠶⢶⣦⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠖⠒⠒⠛⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣧⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣶⣿⣿⣿⣷⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢀⣠⡤⠶⠖⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣄⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣦⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣷⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠞⠛⠉⠀⣇⠀⡇
⠀⠀⠀⢀⣠⣤⣤⣿⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⣟⠛⠛⢶⡀
⠀⠀⢠⡟⢡⡶⠶⢾⡇⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢰⡏⠀⠀⣀⢠⣌⢳⣄
⠀⠀⢸⡇⢿⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠁⠀⠀⠀⠘⣧⠘⠧⠙⠋⠀⣠⡿⠃
⠀⠀⠀⠳⣌⠙⠶⠶⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣠⡴⠞⠉
⠀⠀⠀⠀⠈⠙⠛⠛⠋⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣦⡀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⠶⠖⠒⠛⠋⠁⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⣹⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠓⠦⢭⣙⣓⣶⠶⠶⠶⠤⠤⠤⠴⠶⢚⣛⣉⡥⠗⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣡⣤⡤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀
⠀⢠⣶⣿⠿⠿⣛⣛⣛⣭⠭⠭⠭⠭⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢛⡛⢛⠿⠿⣿⣶⡄
⠀⣿⣿⠶⣯⣍⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣵⠶⣿⣿
⠀⣿⣿⠀⠀⠈⠉⠩⠙⢛⣛⣛⣒⣲⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣖⣚⣛⣛⣿⠿⠭⠍⠑⠂⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣶⣾⠿⠿⠿⠿⠿⠿⣿⣶⣄
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⢀⣤⣴⣶⣶⣶⣦⣄⠙⢿⣷⡄
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠟⠉⠀⠀⠀⠉⢻⣷⡄⢻⣿⡀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⣿⡇
⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣿⡟
⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⢰⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣠⣿⠏
⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⣴⡿⠋
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣴⣾⡿⠛⢁⣴⣿⠟⠁
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠿⠟⠋⢁⣠⣾⡿⠋⠁
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⡀⢀⣠⣶⡿⠟⠋
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠁
⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠁
⣾⣷⣷⢿⣶⠶⣶⣶⣶⢷⣷⡶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿
⣿⣿⣦⣷⣼⣶⣧⣾⣶⣿⣯⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣂⡤⣤⠤⠄⡀⠀⠀⠄⠂⠀⠒⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠲⠾⠿⣷⣤⣤⣠⣄⣀⣠⣤⣴⣤⣤⣴⡤⠴⠒⠓⢉⠽⡰⢈⣉⡀⡈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢽⠂⠐⠤⣄⣤⢞⣵⣧⠖⠒⠒⠲⠌⢷⣰⣦⡈⢅⠘⠃⠃⢿⠁⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡀⡀⢀⡈⢉⢃⠀⠤⢄⠬⣱⢤⡄⠨⠹⣩⢬⡃⣢⠽⠣⠌⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠓⠠⠨⡉⠳⡏⠁⠤⣄⠦⣄⠳⣀⠀⠘⡟⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢰⠀⡇⠀⡇⠘⣐⢿⣏⡯⢰⠿⠂⠀⠃⠀⡇⠁⠀⡆⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢨⠀⣃⠀⡇⠀⠈⠸⠕⠐⠫⣤⣹⠿⠿⡂⣃⡀⠀⢀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢐⣀⡻⠓⢻⡿⡷⡄⣤⣜⡿⣿⡗⣜⠼⢿⢹⡄⠀⣅⡀⠀⠀⢀⠄
⠀⠀⠀⣀⣠⡤⣤⣤⣵⣦⠀⠀⢾⣿⣱⣷⡄⢱⣯⡇⢈⣿⡉⣻⠳⣗⠒⣴⣦⠀⠀⢿⡴⢶⣚⡫⣔⠦⠤⢄⡀
⠀⣾⣟⣏⠀⠀⡷⢋⡓⠁⢦⡈⢩⣻⣤⣹⠘⠃⣽⣷⠐⠿⠛⣟⣶⣶⡿⠛⠀⢀⠜⠘⠒⠋⠉⢑⡴⠽⠭⣷⡝
⠘⡧⣏⣁⠀⠀⠁⠁⠀⠀⠀⠙⠦⣍⢹⣯⣿⣾⣾⠛⠃⠰⢾⣽⣿⣷⢃⣠⠔⠁⡀⡄⠀⠀⠒⠂⠚⠀⠠⢚⠕
⠀⠈⠓⠒⠦⣴⣆⣤⠤⢐⡂⡀⡀⠀⠰⠮⠟⠉⠙⠀⠀⠻⡝⠛⠿⠟⠋⠃⣤⣄⢀⣖⣨⣷⠴⠖⡢⠕⠊
⠀⠀⠀⠀⠀⠀⠨⠉⡉⠉⠉⠉⠉⠉⠙⠛⠙⠛⠒⠒⠒⠒⠛⠛⠛⠛⠻⠿⠿⠿⠛⣋⡩⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⣀⣀⠠⠭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⣦⡾⢻⡿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣅⡠⠾⣿⣿⡵⠟⠁⠀⠀⠀⠹⣶⠀⠀⠀⣴⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠊⠉⠁⣠⣶⣴⣤⠀⠀⠀⠀⠀⢀⡀⡀⣠⠶⠋⠑⠀⢦⠀⠤⠄⣠⡀⠀⠀⠀⠀⠈⠉⢿⠾⠁⢀⣶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠛⠁⠀⠀⠀⢠⡾⠛⠉⠀⠙⠷⣤⡀⢠⡀⠛⢩⠟⢁⠀⠠⠃⠄⢀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢀⣶⡥⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠁⠘⠊⠉⠀⠀⠀⠀⣿⣷⠀⣸⣿⣿⠀⢀⡴⢊⠿⠋⠠⠀⢠⠆⡴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠀⡀⠀⠀⠀⠀⠀⢀⡈⢻⣦⡼⠋⠀⡠⠞⠀⢀⣿⣿⣤⣿⣿⣿⣗⠁⠀⠁⠀⠀⣠⠐⢋⣠⣾⡿⠟
⠀⠀⠀⠀⠀⣼⣆⡀⠀⠀⠚⢁⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⠈⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⢊⣠⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠙⠛⠿⣿⣧⣰⠋⣶⣄⢀⠀⠀⠀⠀⠀⠀⢠⣟⣁⠀⠀⠀⠸⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣇⣠⣦⠀⢡⣉⠁⢰⣷⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠿⣿⠟⠛⠛⢳⣦⡀⢀⢠⣼⣿⠟⠁⠀⣠⣦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡄⣿⣿⣷⣾⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣄⠀⠀⠀⠀⠘⠿⢾⣿⣿⢁⡀⠀⣄⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠉⢀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠳⣤⡄⠀⠀⠀⠉⠉⠉⠙⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⠠⠀⠊⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⣴⣾⡟⣀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣠⣶⣶⠗⠁⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⢀⠈⠉⠀⠛⠓⠴⢄⡄⠀⠀⠈⠿⣆⣀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣰⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⠿⣧⣶⣦⣨⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠛⠋⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⣴⠟⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠙⠿⠛⣿⣿⣿⣿⣿⣿⣯⠉⠀⠁⠀⠀⢀⠀⠀⢀⣤⣾⣿⡶⠛⠁⠀⠀⠔⠋⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠿⠿⠟⠋⠉⠈⠀⠀⠀⠀⠀⠀⠁⠀⠐⠛⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡏⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢻⡆⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠘⠃⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⣀⣤⡴⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⠈⢻⣆⣤⡤⣤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡿⠖⠲⣦⢹⡄
⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠸⣧⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⡉⠉⠉⠁⠀⠀⠀⢰⡟⠀⣀⡴⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣾⠋⡿⠙⡇⠀⠀⠀⠀⢀⣿⠷⢛⣩⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠠⠤⠖⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣠⡿⠷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠟⠳⣤⣀⡀⠀⠛⠛⠛⠛⠃⣀⣠⡶⠛⠶⠦⣤⡀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣄⣀⣀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁⣀⣀⣀⣤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⠤⠤⠤
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⡴⠀⠀⠀⢸⠀⢀⣀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢴⠀⣶⢸⡉⠉⡿⠀⠀⠀⣿⢙⡆⢸⣧⠀⠀⠀⢸⠀⣿⡿⠀⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠧⣤⣤⠴⠃⠸⠶⠛⠈⣷⠞⠁⠀⠀⠀⠘⠛⠀⠘⡇⠀⠀⠀⠘⠀⠈⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⣠⣶⣿⠿⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠳⢦⡀
⣼⣿⣿⣏⠀⠀⠀⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡁⠀⠀⢈⣿
⣿⣿⣿⣿⣶⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⣴⣾⣿
⠘⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣶⣶⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠰⠿⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠁⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠛⠛⣿⠀⣼⣧⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⡾⢋⣴⠟⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⣤⠶⠟⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡶⣤⣀⡀⠀⠀⠀⠈⠻⣦⣄⣀⠀⠀⠀⠀⢀⣀⣀⣠⡼⠋⠉⠁⠀⠀⠀⣀⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣉⠙⠛⠶⠶⣦⣤⣤⣉⣙⣛⣛⣛⣛⡛⣉⣉⣁⣀⣀⣤⣤⡶⠞⣻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣉⠉⠉⠉⠉⠛⠛⠋⠉⠉⣉⣉⣠⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁






⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿