⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠢⠖⡧⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣄⡤⢤⣠⠞⠉⠈⣆⣤⣀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⢰⡿⣡⡷⢱⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠀⠀⠻⠶⠂⠀⠉⠀⠘⠒⠚⠛⠛⢻⡇⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⢸⢣⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢈⠌⡰⠓⣁⠞⠀⠀⠀⠀⠀⢰⣴⡏⢉⣠⣴⣶⣿⡄⠀⠀⣷⡀⠀⠀⢠⣿⣷⣶⣄⡈⢙⡶⡆⠀⠀⠀⠀⠀⠀⡠⢞⡈⡕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠛⠊⠉⠉⠁⠀⠀⠀⠀⠀⢠⡿⣏⣴⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣦⢵⣿⡀⠀⠀⠀⠀⠀⠉⠉⢧⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⢸⣿⠋⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢻⣿⣿⣯⠜⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢿⡿ ⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣏⡇⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢸⣿⣷⠀⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⣤⣄⣹⣿⡇⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣿⣧⠎⢿⣷⣿⣿⣿⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣷⣾⣿⣿⡿⠁⢘⣿⠙⢿⣦⡀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⠇⢣⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⢻⡧⠀⠀⠙⠿⠿⢿⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⣿⡿⠿⠿⠋⠀⠀⣼⡟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢠⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⠒⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⡀⠀⠀⡐⠖⢂⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡟⢛⠈⣔⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢳⣤⣬⣬⣴⣦⣀⣀⣀⣀⣶⣤⣭⣥⣤⡶⠛⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣀⡸⣹⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢛⡿⡃⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢶⣤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡴⠲⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⢡⡳⣄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠓⠚⣇⣧⠀⠀⠀⣠⡀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠫⣧⡐⠛⠉⠟⠁⠀　　⠈⠻⠈⠂⣵⡟⢁⣼⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢓⠄⣚⢭⠂⠵⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⡼⠀⠀⠀⠛⠃⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⠙⢿⣆⠀⢹⠀⠿⠷ ⠺⠷⠀⡇⢀⣼⢿⣤⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡀⠀⠀⠎⡜⠀⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆⢈⣙⠛⠛⠿⠿⠿⢦⡨⠿⡳⡀⠻⣦⣤⣤⣤⣴⠟⣫⢋⡼⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⢺⣷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠎⢩⡜⠁⠀⠀⠀⠀⠈⠙⢲⡙⢿⡿⣫⠟⣽⡙⢿⡿⢋⣟⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠞⢇⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣴⣿⣿⠛⣿⣷⣴⣿⣿⠀⢠⣤⡄⠀⢸⣉⡺⠣⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠸⠇⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⠀⠈⠛⠁⠀⠀⢃⢁⣿⠦⠜⠀⢰⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣧⡿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⠇⡌⠀⠀⢀⣿⣾⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⣯⣾⡾⠃⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠙⠿⠿⠿⠋⠀⠀⠀⠀⣠⠮⣜⠀⠀⢰⣏⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⡲⣀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢛⡃⠀⠀⣿⠛⢛⠛⡇⠀⠀⠀⠀⣀⣠⢴⣄⣨⣯⣀⣀⣀⣀⣠⣤⣤⣤⣠⡤⠒⢉⡮⠊⢉⡇⢠⡟⡛⢏⣽⣧⠀⠀⠀⠀⠀⠀⣀⣉⢹⣭⠇⠀⠀
⠀⠀⠀⢠⢤⣴⣦⠾⢒⡏⣾⣲⣶⣏⣛⣘⣃⣿⣿⡿⠿⢿⡛⢻⡛⠛⠛⢛⣓⠓⠒⠒⡶⠚⠻⠛⠛⠛⠉⠿⠿⠿⠿⠿⢿⣧⣏⣘⣯⣿⣤⣄⣀⣠⠤⣲⢙⡟⢣⣤
⠀⢀⡴⣾⣿⣷⣼⣗⡿⠯⠓⠚⠉⠁⠀⠀⠙⢤⡀⠀⠉⠙⠛⠚⠢⠤⢤⣘⣛⠀⠾⠇⠀⣀⠀⠀⠀⠀⣀⣠⡤⠤⠐⠚⢫⡝⠃⠀⠈⠉⠀⠈⠉⠛⠻⣷⣧⣠⡼⣦⣻
⠀⣿⢡⠿⢧⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣿⢶⣿⡷⢿⣉⣉⡉⠀⠀⠀⢀⡴⠋⠲⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⣹⢿⣤⣽⡇⠀⠀
⢠⣻⡽⢿⡟⠷⣤⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⣸⣷⣷⣍⠶⣤⠴⠒⣚⣭⣽⣿⣶⣿⣿⣾⣿⣷⣾⣦⡈⠉⠉⠓⡶⠯⠤⣀⣀⡀⠈⣁⣀⡀⠀⠀⠀⠀⢀⣸⣿⣿⠿⠀
⢀⣿⣉⡬⠙⢿⣽⠀⠀⠀⣀⣀⡠⠶⠶⠒⠊⠉⠉⠁⠀⠀⢈⡑⢄⠙⠻⠿⠿⠿⠿⠿⠿⣿⢿⣿⢟⡃⠀⣠⠎⠁⠀⠀⡀⣀⡉⠉⠛⠛⠷⠤⣀⣀⡀⢸⡻⠴⡅⡠⠀
⠈⠛⠶⣦⣴⣺⣋⡒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠚⠳⢶⡒⠒⠒⢲⢖⣲⣻⠛⠛⠈⡱⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣗⣋⣉⠀
⠀⠀⠀⠀⠈⠙⠓⠺⢿⣖⣢⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣠⣞⣻⣟⣳⣄⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣒⣻⣥⠞⠋⠉⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠛⠓⠦⣭⣽⣗⣒⣒⣠⠤⠤⠤⠤⠤⠤⠤⣀⣿⡟⠃⠘⢻⣿⡄⣠⠤⠤⠤⠤⠤⠤⠄⡐⢒⣒⣿⣯⣳⡿⠯⣝⣉⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⢿⣦⣤⣤⣴⡿⠖⠒⠒⠚⠙⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀