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