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