⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⡘⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⢠⣡⡤⠐⢤⠤⣠⠃⡞⠀⠀⠀⠀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⣷⣤⣿⣿⡿⢶⣌⣳⠏⢰⣧⠀⠀⠀⣰⠳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣣⣸⡿⣿⣏⣐⡿⠯⣽⡶⣽⠻⡆⠀⢠⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⢀⣤⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⣽⣿⣅⡡⣴⠟⣼⡿⠃⣠⠟⢀⡏⢠⠇⠀⠀⠀⠀⠀⠀⡰⠊⣹⣾⠀⢸⣼⡇⣧⣀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⣯⡋⣹⣁⣴⣋⣠⣞⠋⢀⡼⠉⡿⠀⠀⠀⠀⠀⠀⣰⠁⠀⡇⡘⢀⣼⣿⣇⢉⠟⣦⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⣤⠄⣀⠀⠀⣱⣬⣿⣿⣿⣟⣀⣼⣿⣿⣿⣿⣷⣿⡿⠔⠒⢒⠲⢄⣰⠇⠀⠀⡇⠇⡘⣾⡁⣿⣾⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⢧⢤⡋⠀⣩⣭⣟⣽⣿⣸⣿⣿⠓⣹⢿⣿⣷⣻⣽⣛⣿⢸⣀⣀⠠⢄⣳⠶⠟⠀⠀⢸⢀⠁⡗⣿⣿⣿⣿⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⣂⣼⣈⣸⣿⣿⣿⣵⣾⠟⢉⣽⣵⣿⣿⣿⠟⣫⣵⡿⣿⣟⡉⠀⣾⠛⡅⠀⢀⠁⠀⠀⢸⠊⠀⠳⣹⣿⣿⡿⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⣆⣸⣿⣿⣿⠋⢹⠓⣊⣿⣿⣿⣿⡭⢶⣿⣿⣻⣽⣿⣦⡼⠋⠉⡹⢲⢢⠎⠀⠀⠀⡀⢠⣾⣿⠟⠛⠿⣷⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣧⣄⣸⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⠿⣿⡿⣧⣤⣿⣻⣿⣧⡤⠜⢃⣿⣿⣤⠤⣧⡜⠣⡘⠿⡿⠀⠘⣸⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢩⣵⠬⣛⡇⢈⣿⣿⢼⣰⠯⢿⢾⣍⣀⣀⣹⣝⣿⣽⡷⣿⣻⣿⡿⢻⣿⣟⣇⠹⡿⣿⣿⣆⠑⣀⣹⠲⣾⢿⡟⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⡏⠛⠀⢨⣿⠛⢿⣿⣧⣆⣠⠾⣛⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣿⡀⣯⠙⢳⣉⣍⣿⣇⡞⠈⡀⠙⢹⣛⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣷⡽⣇⡴⠿⡿⠁⠀⠀⠙⢿⡘⡅⠀⢉⣬⠏⠉⣺⣿⣽⣿⡟⠋⠁⠀⠀⠀⢻⡹⣷⡈⠀⢿⣿⣿⡿⠀⠀⡇⠀⢸⢸⢸⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣟⣿⡖⠉⠻⣿⣿⣾⡶⠃⠀⠀⠀⢠⢾⢾⣧⢖⡫⠷⠖⣾⣿⣿⣿⠿⣓⢲⠀⠀⠀⠀⠀⢱⢾⣿⣆⢸⣿⣿⡇⠀⠀⠣⠤⡼⣸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣟⡿⡽⠑⠔⢒⣽⣻⣿⠃⠀⠀⠀⣀⠎⣸⣾⡋⢋⣠⠜⣡⣾⣿⢻⣿⠃⠘⠙⠣⢄⠀⠀⠀⢠⡿⠋⣙⣮⣿⣷⣧⡔⠆⠀⠀⠀⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⣷⢯⣿⡇⠈⣴⠋⢻⡿⠋⠀⢀⣔⣈⣤⠾⣿⣿⣴⣿⣩⡶⠿⣽⣾⣿⠷⣦⣄⣀⣠⠊⢱⣆⠀⡟⢲⠞⠁⠀⠁⡀⣿⡆⠀⠀⠀⣼⣿⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣟⣿⣿⣷⣼⠛⢙⡟⠀⠀⠠⢫⣿⡿⢻⠀⠈⡟⠨⠍⠏⠀⠈⢹⡿⠋⣰⣿⣿⠃⢀⠔⡿⣿⡄⡗⢸⡄⠀⠙⡆⠁⢸⣱⠀⠀⡜⣼⡿⣄⠀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⣽⣿⣿⣟⣹⣿⡿⠀⠀⠀⡟⣿⣿⠓⢚⠀⢀⣝⠀⡦⣦⡀⠀⡼⡏⠀⣿⣿⡇⠈⠀⡆⢻⣾⡇⡽⠐⡆⠀⠀⡇⠀⠸⣏⣑⣾⣶⡿⢠⡏⠀
⠀⠀⠀⠀⣠⠴⠿⣟⣾⣿⣽⣿⣿⣼⡇⣿⣿⣷⣶⡆⠀⡇⣿⣿⡄⠀⡦⠊⢾⡄⢱⡿⠈⢲⣧⣇⠀⣿⣿⣇⠀⠀⡇⢸⢹⣧⣟⠐⣇⠀⠀⣇⠀⠀⠏⠛⠻⣞⡇⢸⠀⠀
⠀⠀⠀⣸⢉⡿⣿⣿⣿⣿⣾⢷⣿⣿⣯⣿⣿⢟⣿⠇⠀⣶⢸⣿⡇⠀⢹⡄⠈⡾⣮⣷⠀⣼⢿⣿⠀⢸⣿⣿⠀⢀⣻⢺⡿⣿⡿⠀⡇⠑⡄⢋⢢⠀⠇⠀⠀⣼⣠⡟⠀
⠀⠀⠀⠘⠬⣛⣷⣻⣿⢻⣟⣿⣟⡁⣉⣩⣿⣾⡏⠀⢸⢻⡆⣿⣷⠀⢸⡿⠒⢷⠈⣹⢶⡟⢺⡇⣧⠘⣿⣿⡄⡸⣅⠀⢻⣿⡀⢢⠟⠀⠈⣳⢨⠀⢸⠀⢰⣿⢸⣣⠀
⠀⠀⠀⠀⣀⣴⢿⣿⡟⢻⣿⣿⣿⣿⡿⠟⠟⠋⠀⠀⣾⠘⢿⣿⣿⣦⠟⣿⣄⠈⠳⠟⠰⡇⢸⣿⠸⣦⣿⣿⡧⢣⣾⣦⠈⣻⡕⡇⠀⠀⡎⠀⢹⠀⠀⢢⣏⣯⣾⡅⠀
⠀⠀⠀⢸⣿⢧⣿⣿⣿⡾⣟⣿⣿⣿⡄⠀⠀⠀⠀⠠⡏⠀⠀⠉⠉⡎⠀⠁⢸⠄⢲⠀⣸⠇⠀⡏⠀⠀⠀⠀⠀⢸⠀⢸⠀⢹⡇⡇⠀⠀⡇⠀⢸⣧⠚⠙⣶⡿⣽⠃⠀⣿⠀
⠀⠀⢀⣾⡿⢋⠇⢠⣾⣿⣽⣟⣾⡽⠃⠀⠀⠀⠀⠠⡇⠀⠀⠀⢠⠃⠀⠐⣼⠀⢨⠟⣽⠀⠀⡇⠀⠀⠀⠀⠀⡝⠠⢸⠀⢺⣇⢧⠀⠀⡇⠘⠟⠳⣆⣴⠾⣯⡽⠀⢸⡟⠀
⠀⠀⣸⣿⠀⡞⣰⣿⣿⣿⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠸⠀⢀⡴⣏⢀⡏⢠⠇⠀⠀⢳⡀⠀⠀⠀⠀⡇⣰⠋⣇⣟⣧⢸⠀⠀⡇⠀⠀⠀⣼⠁⠀⠀⡇⠀⣿⠇⠀⠀
⠀⠀⣿⣇⣤⣷⣿⣿⣿⣟⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⡏⠀⣽⣾⣅⡾⠁⠀⠀⠘⣧⠀⠀⠀⠀⣧⡏⢀⣿⣿⣷⢸⠀⠀⡇⠀⠀⢀⣿⡀⠀⢸⠃⢸⡿⠀⠀
⠀⠀⢻⡿⢁⣿⣿⡾⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣏⣯⡿⠀⣾⠋⢉⢿⡆⠀⠀⠀⠀⢫⣣⣶⣶⣶⣼⠃⣼⡇⠸⣿⢸⠀⠀⡇⢀⡴⢋⡇⢳⡀⡿⠀⣼⠇⠀⠀
⠀⠀⣼⠃⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⡿⣥⠞⢹⠀⢸⣈⣷⠀⠀⠀⠀⠀⠣⣿⣿⡿⠿⡼⠁⢧⠀⢿⡜⡆⠀⡷⠋⠀⣸⣇⠀⢹⡇⢰⡿⠀⠀⠀
⠀⢠⡇⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣘⣣⣈⣢⣌⣳⣤⣿⡁⠀⠀⠀⠀⠀⠀⠈⠹⣿⡐⠒⠁⣸⣷⢻⣇⣧⠈⡇⣠⣾⠿⣿⣆⢸⠳⣼⡇⠀⠀⠀
⠀⡾⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠒⠒⠒⠉⠿⢷⣣⣃⢹⠀⠀⠀⠀⠀⠀⠀⢸⡋⠍⠨⠥⢞⢻⣬⣿⡇⢀⡟⠁⣾⢸⣿⠈⣿⠀⣿⠀⠀⠀⠀
⣾⣡⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢸⠀⠀⣼⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⠀⣼⢻⡷⠁⣏⠀⣿⢸⡏⢰⡏⣻⡇⠀⠀⠀⠀⠀⠀
⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣤⣤⡇⣼⠁⢠⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡆⢢⠀⠀⢨⣤⡇⢸⢻⠀⡏⠀⣿⣿⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢻⣀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⢃⠀⠟⢿⣿⡿⣇⢸⣿⠀⡝⠀⣿⣿⠃⣏⢸⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠊⠌⠙⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡸⢼⢱⣧⠚⣶⡟⣄⡹⢸⢳⣿⢀⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠈⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣾⣼⠀⣿⣇⠀⣷⣼⢸⣟⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣤⣷⣿⠀⣠⣼⣻⡹⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⢻⣏⡆⣿⡟⣆⡽⠀⠈⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠋⠉⣉⠛⠯⠜⠁⠀⣿⠇⣠⣾⡇⠀⠀⠀⠀⠀⠀⡿⣿⣷⣘⠿⠧⢿⣷⠟⡇⠀⠀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢈⢶⣦⠀⢸⡟⣴⣷⡀⡇⠀⠀⠀⠀⠀⠀⡇⠹⡫⠀⠀⠐⠒⠫⣄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⢗⠒⢤⡪⡈⠉⠳⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣧⠾⠀⠀⠀⣀⠀⣅⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠉⢀⢀⣤⣿⣮⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀�