⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣡⣴⣿⣿⣿⢿⣿⣄⠀⠀⣀⠤⠔⠒⠚⠉⠉⠉⠉⠙⠲⡖⡒⠦⣄⡀⠀⢀⣠⣾⣿⣦⡀⠄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣾⣿⠾⢋⠴⠻⣿⡿⣿⣷⡋⢁⠤⠠⠐⠒⠀⠀⠠⠄⠀⠀⠹⠤⠃⣤⡉⠲⣿⡿⡟⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠹⣯⣿⣧⡤⠏⢰⡡⠘⢿⣟⡿⠛⠁⡀⠀⠐⠀⠀⠀⢀⠔⠉⠙⡂⠀⠀⠀⠈⠳⢤⡑⢦⣈⢳⣿⣿⡄⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠙⢿⠟⠀⠀⠀⠑⠉⡼⠋⣀⡔⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⢻⣿⣿⣆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⡂⠈⠀⡎⠀⠀⠀⠀⠀⡔⠡⣊⠌⠀⠀⠀⠀⠔⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠈⢀⠀⠀⠀⠀⢻⠸⣇⣿⣿⡿⠃⡐⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠎⠀⡀⡸⠀⠀⠀⠀⢀⠎⠀⡳⠋⠀⠀⠀⠀⡀⡀⠀⢀⣾⠀⠐⠄⢸⠘⡄⠀⠀⠀⢄⠀⠀⠀⠘⣆⢿⣿⡇⠠⠊⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠤⠀⠀⢀⢀⡇⢠⡀⠀⢀⠎⠀⢠⠁⠀⠀⠀⠀⡰⠀⠐⣲⠟⢹⠉⠀⢀⢸⡄⠹⣦⣄⡁⠈⠄⠀⠀⠀⣿⡟⡩⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡘⠄⡧⣔⣁⣀⣴⡇⡴⠁⠀⠀⢠⣧⣔⣮⣵⣶⡤⡆⢰⠀⠌⣿⣄⣹⣦⣘⠩⢞⠀⠀⠀⢸⡗⠠⡇⠸⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡀⢛⠇⠀⠀⠘⣿⣿⠃⠀⠀⢿⣿⣿⢿⡿⠛⠻⢿⣿⢜⡀⠀⢠⠛⣧⠻⣿⣧⣼⠀⠀⠠⢸⡇⢃⢧⠀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡀⡏⠀⠀⠀⠀⢿⣿⠀⠀⠀⢀⡿⣅⠸⠀⢯⡗⠸⠏⠉⠑⢤⠈⢣⣼⣻⠻⡿⣻⣷⣤⠀⡆⡧⢌⣸⠀⠉⢦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡼⡐⡵⠁⠀⠀⠀⠈⣿⠛⠆⠂⠀⠘⡅⠀⠙⠂⠀⠀⠊⠀⠀⠀⠀⠉⠒⢛⠉⢑⠃⣠⣏⠀⠀⡇⢱⡂⠤⢇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⢡⠞⠀⠀⠀⠀⠀⢰⢳⠀⢇⠀⠀⣶⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠉⠁⣿⠀⠁⡿⣄⣙⣤⡈⢆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡱⠋⠀⠀⠀⠀⠀⠀⣧⢺⠀⠘⢆⠀⢸⣏⠙⠀⠀⠀⠀⠀⡏⠉⠑⠒⣦⡄⠀⠀⠀⢀⣴⠻⠀⢠⡇⡔⢠⠖⠑⡬⢶⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡞⠊⠀⠀⠀⠀⠀⠀⠀⢰⢸⢸⡇⠀⠘⢦⠘⡟⢿⣦⣀⠀⠀⠀⠈⠐⠂⠐⠉⠀⢀⣀⠖⢉⡇⡇⠀⡞⣇⡘⠤⡄⠀⢘⠀⡜⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣸⢸⠇⣀⣀⡤⠗⢿⡀⢉⡿⢿⣶⣤⠄⡀⠀⣀⠤⠒⠉⠁⣠⢮⢼⣀⠞⡇⡧⢌⠢⠰⡀⠘⡄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠜⠛⠉⠉⢇⠘⡼⡀⠀⠱⡞⢀⠎⣠⠏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠘⡇⡇⢆⠱⢡⠡⠀⢱⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠰⡤⠤⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⣖⠲⡙⢄⠜⠀⡜⡹⡍⢆⣺⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⢡⠊⠴⡁⠎⣅⠀⠇⢁⠀
⠀⠀⠀⠀⠀⠈⠐⠌⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢀⣿⠀⢹⡎⠀⠀⠉⢳⣸⣌⠃⠸⡝⢦⡀⠀⠀⠀⠀⠀⠀⢠⠏⡮⠤⡙⠤⡑⠌⡟⡄⠘⠀⠄
⠀⠀⠀⠀⠀⠐⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⠃⠀⠀⠀⠀⠀⡔⣚⣹⠀⠀⠀⠀⠀⠀⠀⢹⡀⣀⣠⡟⠤⢫⠄⠀⣀⠀⢀⡠⢋⡼⢁⡒⢡⠂⡕⢪⡇⠸⠀⡆⠀
⠀⠀⠀⠀⠀⡰⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡞⢸⠀⠀⠀⠀⠀⣸⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⠇⠙⠋⠹⡌⠹⡔⠀⠈⠛⠓⠚⡟⠁⠀⠘⠠⠃⠈⢆⡇⠀⡇⢰⠀
⠀⠀⠀⣠⠚⠆⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⣠⠚⢤⠯⠀⣼⠀⠀⠀⡠⢴⡿⡄⠸⡟⢀⠀⠀⠀⠀⣠⠾⡇⠀⠀⠀⠀⢻⠐⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘⣰⠀⠀⡇⢸⠀
⠀⣠⠚⡡⠋⠀⠀⠀⠀⠀⠀⠀⡜⡇⠀⡴⢁⢪⠗⠁⢰⠉⠀⠂⢉⣤⣿⣹⢷⡀⠙⢦⡈⠐⣶⣾⣷⢘⡇⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢼⠀⢰⠁⡜
⡠⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧⢰⢁⣮⡫⠀⠀⢠⣿⣶⣿⣿⣟⡶⢯⣞⡷⣤⡀⠙⠚⠿⠟⠛⡺⢃⠀⠀⠀⢀⠞⡀⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠏⡶⢁
⢡⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢊⠻⢦⠇⠀⠀⠀⢸⣿⣿⣿⡿⣼⢽⡻⣼⠳⢯⣟⠷⣶⣤⡶⠊⢀⢫⠀⠀⡠⠛⠀⠒⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡵⠊⢀⠐
⢠⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠐⡄⡔⢡⡏⠀⠀⠀⠀⣼⣿⣿⣿⡟⡞⣧⡟⠁⠀⠀⢩⣿⣷⣿⠀⠀⠁⢸⠀⡞⢣⠀⠀⠀⢳⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠊⠊⠀⠀
⠂⠀⠀⠀⠀⠀⠀⢠⠎⢡⠈⠑⠀⠑⡞⠄⠀⠀⠀⡴⣹⣿⣿⣿⣟⣳⡿⠁⠀⠄⣠⣾⡽⡛⣉⠀⠀⠀⢸⣸⣷⣌⠑⠤⢀⣼⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡰⡟⡈⢄⡈⠡⢄⢸⡑⢤⣤⢔⣊⣴⢿⣿⣿⣿⣯⢯⡀⠀⣠⣼⣿⡏⢰⠟⠛⠀⠀⠀⠸⣹⣿⡿⣷⣶⣤⣼⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⠎⠀⠈⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⢩⡇⡱⢌⡐⡧⢌⢸⠃⠀⠀⢠⠔⠋⠁⢿⣞⡽⣏⡿⣹⢷⣻⣿⣿⠅⢜⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣻⣟⣿⡄⠀⠀⠈⢣⡀⠀⡠⢔⡏⠀⠀⠀⡇⠄⠀
⠀⠀⠀⠀⠀⣿⠠⣇⡑⢢⠐⣿⡰⠘⢧⣀⣀⣠⣆⠀⠀⠘⣿⡞⣽⢾⣹⢧⡗⣯⣾⠀⠈⣂⠀⠀⠀⠀⡠⠊⠸⣿⣟⣿⣻⣿⡇⠀⠀⠀⠀⠈⠳⣕⣸⠀⠀⠀⠀⢇⠈⠀⠀
⠄⠀⠀⠀⠀⡟⢦⣻⡄⠣⢌⢻⠳⣉⣴⣿⣿⣿⡿⣦⡀⠀⠘⠿⢧⡿⠧⠿⠟⠋⣠⣴⠿⣿⣷⡄⣠⠊⠀⠀⠀⣹⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⢸⠓⢒
⠌⠠⣀⠀⠀⣧⠃⢍⢋⠱⣈⣶⣿⣿⣿⣿⡿⣯⢽⡳⢷⣄⠀⠀⠀⠀⢀⣤⢶⣿⡻⣼⣛⣿⣿⡟⣯⢧⠀⠀⣰⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡄⠀⠈⢷⠞⠋⡠⡀⠈ ⢸⡌⢢⠘⣸⠘⢿⣿⣯⣷⢿⡝⣾⢳⣛⣟⣾⣷⣤⡴⡾⣏⡟⣾⢷⡻⣵⢫⣿⣿⣝⡾⣽⣷⣴⡟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢄⠰
⠑⡀⠀⠀⠀⠀⢷⡁⠎⡄⠓⠬⣿⢿⣯⡳⣟⣭⢟⡽⣾⡿⣞⢶⡻⣵⢯⣻⢯⡗⣟⡾⣝⣿⡿⣼⢳⢧⣿⣳⡻⣽⣿⡿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣈⠺⣦⢦⠀