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