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    '        ‘               ¯¯¯x                                                                                      '  ‘                ¯¯¯°                                                          ‘            '               

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣬⢿⣿⣦⡄⠀⠙⢷⣆⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣻⣿⣧⡀⠀⠀⠙⠿⣶⣄⠀⠐⠀⠢⠨⠨⠈⡉⡙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣷⡀⠀⠀⠀⠘⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠂⠢⢈⠙⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠦⡙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣎⠹⢿⣿⣦⡀⠀⠀⠸⣿⡹⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠐⢈⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⢹⣿⡄⠈⠙⢿⣷⣆⡀⠀⢿⣇⠂⠝⠿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠌⢿⡿⣷⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⠛⢿⣶⣄⡀⠀⠀⢸⣿⡿⣷⣄⡀⠉⠻⣷⣆⣸⣿⡆⠀⠑⠨⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡻⣎⠹⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠲⠤⡀⠉⠻⣿⣷⣶⣾⡿⠃⠌⡛⢿⣦⣄⠈⠙⢿⣿⣷⣦⡀⠀⠑⢈⠻⣿⣯⡲⢤⡀⠀⠀⠀⠀⠀⠐⢈⠷⣄⢻⣿⡿⣷⣤⣀⠀⠠⠤⠤⠤⠤⠤⢠⣀⣀⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠑⢆⣀⠙⠿⣷⣦⣄⠀⠈⠠⢻⣿⣷⣦⣀⠙⣿⢽⣻⣷⣄⡀⠐⢈⠻⣷⣆⠙⣆⠀⠀⠀⣀⣠⣀⣁⣙⣾⣿⣧⠈⠛⢿⣷⣦⣀⠀⠀⠈⠈⠐⠐⠩⢉⢛⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠳⣥⡨⠙⠿⣿⣦⣄⠀⠝⢿⣟⠻⢿⣾⣇⡈⠙⠿⢷⣦⣀⡐⡘⢿⣧⡈⢇⠀⠈⠀⠀⠀⠩⡏⠉⠛⣿⣷⣶⣤⣈⡹⠿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠈⠐⠍⡳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡀⠀⠘⢷⣮⡐⠨⢙⠿⣷⣦⣌⠻⣷⡀⠉⢛⠛⠷⠆⠈⠙⠿⣷⣦⣡⢻⣷⣸⠆⠀⠀⠀⠀⠀⢻⡀⠀⠻⣿⣎⣉⠛⢿⣿⣦⣍⡻⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⢊⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡴⠁⠀⣀⠀⢀⠙⢿⣶⣄⠁⠌⡙⠿⣿⣿⣿⣆⠈⢲⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⡟⠁⠀⠀⠀⠀⠠⢾⣷⣄⠀⢹⣿⡎⠝⡲⢽⣿⡻⠿⣷⣶⣭⡙⡻⠿⠶⣶⣶⣶⣶⠿⢃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣸⠁⠀⢸⡿⣯⣉⠛⠮⣿⣿⣷⣆⡀⠑⢈⠙⠿⣿⣦⣄⡙⢿⣶⣤⡀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠀⢸⠟⣿⣷⣦⣽⣿⡄⠀⠡⢙⢿⣖⠨⢙⠻⣿⣶⣅⡁⠀⠂⠑⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⢰⣷⡀⠀⢸⣷⠈⠻⣷⣆⡈⢟⢿⣿⣿⣦⣄⠈⠐⠨⠹⢿⣿⣶⣌⡛⠿⣦⣄⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⡟⠈⠜⢿⡿⣝⢿⣿⣄⡀⠀⠘⣿⣇⠀⠈⠐⢙⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⡀⠙⢿⣶⣼⣿⡀⠀⠈⠻⣿⣆⡃⠙⢿⣿⡻⢷⣦⣀⡈⠐⢈⠙⠻⣿⣷⣬⣉⠙⠁⠀⠀⠀⠀⠀⡀⠐⠊⠀⠀⠀⠡⠻⣎⠱⢿⣿⡿⣷⣦⣸⣿⣇⠀⠀⠀⠈⠨⠹⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢿⣿⡔⠠⢙⠻⣿⣧⣀⠀⠀⠘⢿⣿⣄⠀⠙⢿⣤⠈⢉⡛⠛⠲⠞⠖⠖⠟⡻⠿⣷⣶⣄⣀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠁⢚⢷⡄⠹⣿⣌⠙⠿⣿⣿⣆⠀⠀⠀⠀⠀⠁⠝⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⢻⣿⣦⠀⠡⢈⢻⣿⣷⣄⠀⠀⠹⣿⣦⠀⠈⠹⣷⣄⠉⠳⣦⣀⠀⠀⠀⠀⠁⠊⠍⠻⠿⡷⣶⣦⡾⠃⠀⠀⣀⠀⠀⠀⠀⠂⡙⢦⣈⠛⣷⣤⡈⢿⣿⣿⣶⣄⡀⠀⠀⠀⠐⠩⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠻⣿⣷⡀⠀⢂⠹⣿⣿⣿⣤⡠⠼⣿⣧⠀⠀⠈⢿⣧⡀⠈⢻⣿⣦⣬⡱⢒⠦⠤⣄⣌⣀⣁⣀⣀⣄⡀⠀⢀⡑⠦⣄⣀⠀⠈⠐⢉⠳⢤⣙⠿⣿⣿⠘⣏⠻⢿⣶⣄⡀⠀⠀⠡⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠙⣿⣿⣆⠀⠡⠹⣷⡉⠻⣿⣦⣿⡿⣿⡶⣄⡀⠹⣷⡄⠀⠉⠉⠻⢿⣷⣮⣈⡐⠀⠍⠍⢉⢻⢿⣿⣶⣬⣷⣆⠈⠛⠿⣶⣤⣀⠈⠐⠠⢉⢈⢉⢩⣀⠈⠐⢉⠻⢿⣷⣤⡀⠐⡘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⢿⣿⣷⡀⠐⠌⢿⣦⠈⠻⣿⣿⣾⣷⣌⠿⣦⡘⢿⣆⠀⠀⠀⠀⠙⣿⡿⠿⣶⣦⣀⡀⠀⠀⠉⠛⠿⣿⣿⣧⣀⠀⠈⠉⠻⢷⣦⣄⣀⡀⠀⠀⠉⠳⣄⡀⠈⠐⢈⠛⠿⣷⣦⡘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠹⣿⣿⣆⡀⠡⢙⢿⣄⡈⢻⣿⣿⡽⣧⡈⠹⢾⣿⣿⠶⢶⣤⣠⣘⢿⣧⡀⠉⠙⠻⠷⠶⣦⣄⣠⣈⠙⢻⣿⣷⣦⡀⠀⠀⠈⠙⠿⣿⣿⣻⢦⣀⠈⢻⣶⢦⣀⡈⠐⢈⢙⠻⢿⣿⣇⡀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠈⢻⣿⣷⣄⠀⡇⠙⠳⣦⡘⢿⣿⣿⣿⣦⠀⠈⢿⣷⣦⣉⠙⠻⠻⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠻⢷⣶⣤⣀⠀⠈⠉⠻⢿⣿⣷⣤⣹⣆⠈⠉⠓⠒⠶⠒⠊⢸⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠙⢿⣿⣷⣅⠀⠀⠀⠉⠳⠻⣿⣷⣝⢿⣤⡈⢿⡎⠙⠿⣶⣄⣀⠹⣿⣿⠿⢷⣶⣤⣀⡈⠙⠳⠶⣤⣜⣿⣖⡐⠨⢉⢛⠿⢶⣦⣤⣀⣉⡙⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠈⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡙⢿⣿⣧⣄⠀⠀⠀⠀⠈⢻⣿⣦⡉⢻⣾⣿⣁⠂⠌⢙⠻⢷⣾⣿⣷⡀⠈⠉⠻⠛⠓⠂⠠⠀⠉⠙⢿⣷⣦⣄⣀⠈⠀⠂⠊⠩⠩⠩⠁⡈⢹⣿⣿⣽⣶⣄⡀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣿⣿⣷⣄⠀⠀⠀⠀⠻⣿⣿⡄⠹⣿⣿⣷⣦⣀⠈⠐⠨⠙⠿⢿⣦⣄⡀⠀⠀⠀⠠⢤⣄⣀⠈⠻⣿⡍⠙⠻⠶⣦⣀⡀⠀⠀⠀⢐⠈⢿⣿⡍⠻⢿⣿⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⠻⢿⣷⣦⡀⠀⠀⠹⣿⡇⠀⠫⠛⠿⠿⣿⣷⣦⣀⠀⠁⠪⣽⣿⣿⢷⣶⣄⣀⠀⠈⠙⠻⢶⣿⣿⣦⠀⠀⠀⠉⠙⠳⢦⣀⠀⢂⠘⣿⣿⡤⢀⠙⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠙⠻⢿⣶⣤⡀⢹⣧⠀⠀⠀⠀⠁⠊⠍⡛⠿⣷⣦⣀⠀⠉⠛⢷⣌⠉⠛⠿⢶⣦⣄⣀⡈⠙⢻⣷⣶⣤⣀⠀⠀⠀⠈⠁⠀⢂⢹⣿⣷⡄⠐⠄⡈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡉⠙⠿⠿⣿⣤⣀⣄⣀⣄⠴⠂⠀⠂⠄⡙⠻⢷⣦⣀⠀⠉⠳⢦⣄⠀⠀⠉⠉⠛⠳⠖⠿⢿⣯⣻⣿⣶⣦⣄⡀⠀⠀⢂⢳⡽⣷⡄⠀⠈⠠⡈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠟⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠄⡉⠛⠷⣦⣄⡀⠈⠉⠓⠒⠊⠈⠀⠀⠁⠂⡙⢷⣆⡀⠻⠽⣻⡿⣶⣤⣈⢷⡹⣷⡄⠀⠀⠐⡀⢻⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠌⡉⠛⠷⠶⢤⣤⣀⣀⡤⠤⠀⠀⠀⠂⡙⠷⣅⡢⠠⢉⠓⠿⢽⣻⣷⣿⣿⣄⠀⠀⢐⢀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⡙⠦⣄⡈⠐⠠⠨⢉⠙⠛⣿⣿⠶⠶⢿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠍⠁⠀⠀⠀⠈⠂⡈⢻⣧⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⠀⠀⢐⠬⣑⠄⡀⢄⠨⢱⠰⡲⢡⠀⠰⠐⢌⢐⢀⡀⢀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⡑⣇⡃⠀⢐⢰⢣⠭⣊⡪⠀⠬⡸⠃⠝⠀⢨⣘⢲⢸⠐⣰⢑⠢⢣⢩⢍⠂⠀⢀⢢⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡐⠡⠀⠀⢘⠬⠪⠠⢀⠂⠀⠀⠧⠩⠂⠀⠐⡦⢊⢆⠧⠕⠢⠭⢫⢰⢁⠀⠀⠈⠔⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠨⠈⠀⠀⠀⠅⠐⠀⠨⠈⠀⠀⠑⠀⠂⠀⠈⠆⠅⠣⡑⠅⠱⢁⠈⡎⠀⠀⠀⠀⠅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠁⠀⠀⠈⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⢴⣄⠀⠀⠺⣾⣦⠀⠀⣦⡀⠀⢠⣶⣖⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣄⡈⠳⢦⣀⠀⠙⢿⣦⣝⣷⣄⠀⠙⢿⣆⠀⠈⢿⡄⠀⠀⢺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⢀⠘⠿⣶⣤⣀⢀⣠⣠⠞⠷⣅⠉⠓⠃⢙⣷⣦⣌⠳⣄⠈⠙⢷⡄⡈⠻⣟⢦⣄⢿⡄⠀⠀⣿⣧⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣿⣞⡛⠛⠛⠟⠿⣷⣤⣾⣿⣄⡄⠤⠘⠛⠛⠟⢿⣷⣤⠤⠿⢿⣦⡽⣆⠀⠉⢿⣦⠀⢺⡏⠳⣽⡧⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⢤⣤⣄⣀⢀⣠⣉⣙⣳⡦⡶⣊⠁⠈⠁⠙⠙⠛⢷⡶⡦⣦⣤⣤⢬⠝⠻⣶⣤⣝⣯⣛⢶⣤⣄⣹⣷⣘⣗⣀⣈⢿⣦⣀⠀⢰⣿⡇⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣷⣶⣶⣶⠶⠶⠲⠚⢈⣨⠞⠛⠛⠛⠻⠦⢤⢤⣠⣉⣓⣖⡢⡤⠔⠂⠈⠑⠲⠤⠠⠀⣀⠀⠈⠙⠾⢍⠹⢿⣻⣤⠁⠈⠻⣿⣦⣀⠈⣷⡀⠑⢾⡟⢷⡀⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⣈⣩⠛⠛⣷⣧⣶⣴⣟⣁⣀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢯⣢⡀⠀⠀⠠⣄⠀⠀⠈⣯⣂⠀⠀⢀⢄⠀⠉⢻⣷⣦⡀⠙⢿⣝⠙⢿⣧⣄⣸⣇⠘⣯⣾⡇⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣴⣾⠟⠉⠛⠛⠳⢿⠿⣷⠶⠶⠮⠷⠷⠕⠷⣳⢍⢁⠀⠈⠧⣄⠀⠀⠀⠱⣳⢕⣄⠀⠈⠯⡧⡄⢱⠅⠳⢤⢀⢳⡀⠀⠨⡮⢩⠟⠷⠾⠟⣧⡀⢹⣧⣼⡇⠁⠑⢿⣇⠀⢨⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡈⡈⡈⠙⣛⣿⣷⣶⢾⣯⣤⣰⣧⣠⣠⡴⠓⠀⠀⠀⠈⠫⡳⡓⠲⠔⢜⡵⣢⣄⣀⠈⠳⣌⡹⠑⠇⠽⢺⣪⣳⠀⠀⠁⠓⢧⠀⠨⣺⢮⣳⡀⣳⠀⠈⠛⢶⣽⣿⡗⢦⣀⣼⣿⢦⣼⣿⡇⠠⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣮⣿⣿⣿⡟⠻⠿⠿⣦⣶⣦⣶⠾⠓⠋⠉⠀⠉⠓⠖⡽⣝⢶⢔⣬⣹⣢⢄⣀⣌⡠⡱⠝⡽⡝⡊⠺⡲⡒⡢⡒⠑⢗⣗⢮⡈⢁⣉⢧⠀⣗⠙⡮⠘⢵⠅⠀⢯⠂⠈⠻⣿⠶⠈⢿⣿⡤⣿⠎⣿⡐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣬⣤⣤⣶⢾⠿⣿⣾⣿⣿⠟⠋⠀⢀⢦⢤⢤⠤⠔⠖⠑⠉⡰⠁⠉⠘⠘⢪⠏⠞⢖⡖⡖⠖⠖⠮⠓⠳⢝⡮⣣⠀⠀⠈⠘⡵⣈⠑⠱⢯⢦⢼⢄⢜⡧⣹⣉⠲⣝⠀⢠⡆⣿⣷⣄⠈⣿⣇⣿⠁⠘⣿⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣠⣤⣼⣾⢻⣿⠋⡛⠛⠲⠢⠴⠉⠉⠊⠋⠓⠖⢴⡸⣠⢀⣀⡤⢊⡡⣠⢎⢅⠈⠉⠓⠢⠦⠔⠀⢀⡌⠑⠙⢽⣒⡒⠊⢗⢗⡤⡈⠱⣫⢇⠐⡭⠚⣮⡣⡳⠀⢸⡲⢺⣧⠙⢷⣼⣿⡿⠀⠀⠸⣿⠀⣰⣿⠅⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣶⢶⢿⡟⠋⢩⣗⣴⢟⣡⡾⠃⠀⠀⠀⠀⠀⢀⢀⢴⢤⣔⢧⣺⡪⠯⢞⡎⠃⣭⣈⠈⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠉⠁⠀⠑⢎⡉⠓⢖⢽⡒⣞⣄⢷⢝⡇⡰⡕⡗⡗⢉⢦⡀⠀⢹⣿⣆⠀⢸⣿⢰⡿⣿⠁⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣁⣤⡾⢛⣧⣠⣾⣿⡿⠛⠉⠀⠘⠙⠙⡙⣝⠽⠑⠋⡭⡃⠁⠁⠀⣀⢔⢇⠄⠀⠈⠙⠿⣶⣤⢤⣠⣙⣿⣦⡤⠠⠄⠙⢷⣤⠸⣇⠀⠀⠀⠀⣉⠢⠀⠘⣝⢎⢮⢪⢗⡧⢻⢴⠝⢄⡼⢅⡤⠀⠈⣿⣄⠓⢶⣿⡿⠀⢻⡂⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢶⢾⣿⢿⠿⢿⣿⣿⡿⠁⣼⠁⣀⡠⡤⡔⠖⠙⠈⠀⡨⡤⠋⢠⡣⡠⣔⠮⠞⠕⠁⠀⡀⡀⠀⢀⡏⠙⠷⣦⣄⣠⠞⠙⢷⣴⣄⠈⢯⡳⢿⣧⠀⠀⠈⣿⡆⠀⠀⢱⠌⠙⢽⢵⡋⡪⡇⡠⡓⢝⡮⠀⠀⢀⠘⢿⣦⠐⣿⣿⠤⡘⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠻⢿⢁⣼⢏⡼⠓⠙⠚⠊⠚⡪⡺⡮⣫⠯⢯⡳⠉⠈⠀⢦⣤⣶⠶⠛⠙⠛⠛⡻⠻⢾⣦⣶⣟⣿⣦⣀⠀⠙⠻⣧⡀⣳⣄⡹⣷⠖⢤⣸⣿⡀⠀⠀⠫⣄⠞⢱⡳⠭⣹⡊⠀⣞⡆⠀⣰⣫⠇⠀⠙⢿⣿⠘⢧⡀⣺⣇⠀⢀⣼⡟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣀⣤⣴⠾⣛⣽⠋⠀⠠⣿⣿⡟⠋⠀⠀⣀⡠⡤⡺⣝⢽⠉⠀⢠⡺⢠⠀⠀⠀⠀⡙⠙⢛⢓⢖⣦⣴⣷⣾⣋⣁⡀⡈⠉⠙⠛⢿⣟⣙⠋⢿⣯⣳⡄⠺⣧⡀⠀⣿⣷⡀⠀⠀⠈⠀⢘⡮⡵⠁⠑⡼⣸⡪⡼⠡⡎⡄⠀⠀⠰⣿⡀⠈⢷⣿⠻⣴⣿⡏⠀⣀⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠘⣉⣴⢾⢯⢀⣠⣴⣾⡿⠁⠀⠀⠐⠋⠊⢈⡠⠎⣜⡅⢀⣔⢯⠎⠃⠀⢀⢠⣞⣠⡾⣿⠿⢛⣻⡫⠕⠉⠈⠉⠉⠉⠉⠀⠀⠀⠈⠛⠳⢤⠙⣟⠿⣄⣹⣧⠀⣻⠌⢳⣶⡇⠀⠀⠠⢯⢧⠀⢀⠏⡶⡝⠀⣸⡸⠅⠀⠀⢘⣿⡅⠀⢸⣿⣾⣫⡟⢀⣴⡟⠁⠀⠀⠀
⠀⠀⠀⠀⠘⠟⠟⠟⠟⠛⣫⣿⡿⢿⣿⠀⠀⠀⢀⢠⠔⢃⢡⢼⢕⣖⠗⡕⡸⠀⠀⣀⣼⣟⣩⣤⣴⣧⣤⣞⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⡝⢻⣧⢿⣥⣤⢾⢧⠄⠂⢀⡯⡑⢇⡸⣰⠋⠂⢀⢞⠎⠀⡰⡏⠠⣿⣷⠀⣿⡿⠁⢹⣷⣾⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⡿⣠⣿⡿⣠⣔⣖⠽⠵⡚⡹⢕⢥⡳⠁⢨⢇⢏⣰⣾⡿⠙⣉⣩⣭⡿⢋⡄⡦⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢹⣾⠂⠀⣿⠁⠀⣼⡿⢹⡢⠀⣹⣺⠀⠀⢼⡝⢀⣖⡏⣀⠀⠻⣿⣾⣿⢽⡄⣸⡟⠈⠀⠀⣀⠀⠀⠀
⠀⠀⠀⠀⣀⣴⡾⣿⠟⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⣗⠽⠉⢒⡇⠀⡮⠃⠀⠉⢠⣟⣨⡿⢋⡽⢗⡽⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡠⡿⢀⣼⣿⡇⠀⠈⠀⣗⠸⡀⠀⣗⢥⠃⣗⢥⠃⠀⠀⣸⡟⣿⣿⣿⡟⠀⢀⣠⣾⠿⠋⠀⠀
⠀⠀⠀⠘⠋⣩⣼⢋⣦⡶⣿⣿⠀⢰⡆⠀⣀⢤⢔⢟⠌⢆⠀⣜⢮⣺⠁⠀⠀⣤⣾⠟⢁⣴⣿⣧⠏⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡌⢿⣿⣿⠏⣿⢰⡖⠀⠨⣪⠀⡃⣰⡳⠁⠠⣳⠋⠀⠀⢀⡟⠨⣿⣿⣟⣠⣴⢿⡿⢣⣄⠀⠀⠀
⠀⠀⠀⢴⡿⡿⠿⠛⢁⣿⣿⠏⠰⣼⡟⠉⠊⠈⢀⡎⠀⢸⢤⡓⡯⠂⢐⠀⠀⣰⣟⣴⢟⣽⣿⡏⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣇⣾⣿⠿⣄⣿⡟⠀⠀⠐⠔⠄⡼⣕⢯⠀⢨⠞⠀⢀⢴⠚⠅⠨⣿⣿⡾⠋⢠⣿⣷⡿⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⠀⣰⣿⠀⠀⠀⢀⡗⣀⠴⣕⡯⡞⡽⡀⡝⠀⣰⣿⡿⠁⢛⡿⢋⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⢀⣌⣿⡛⠦⣤⢀⡠⠝⡹⡸⠙⡄⡮⠁⡤⡳⠁⠀⠀⢨⣿⡏⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣠⣾⢛⡟⠁⢸⣧⡿⣿⠀⠀⡰⡝⠎⠊⠀⣳⠝⠸⣝⢼⠡⣼⠿⠋⠀⢸⡾⠃⣰⣿⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣵⢞⣿⣿⣷⣿⡿⠕⠁⢸⢂⡇⠀⡮⣣⡺⠝⠀⠀⠀⣠⣿⠛⣿⣀⣾⣿⠀⠀⢀⣠⣦⠀⠀
⠀⠀⠀⢀⣴⠿⠁⣸⡇⢀⣾⣿⢸⣿⡄⠸⠁⠀⠀⣤⡺⣊⡀⢐⡽⡁⢨⠆⠀⠀⢠⣿⡇⣼⣿⡟⠀⠠⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⣿⣾⠏⢡⡿⠋⠀⠀⠀⠕⠰⣅⣞⠝⢢⡏⠀⠀⠀⣰⣿⣟⠀⣸⣿⠁⢷⣠⣾⣿⡟⠁⠀⠀
⠀⠀⠀⠙⠋⠀⣰⣿⣰⣿⣫⣿⡿⢙⣧⠀⢀⡠⡮⡃⣸⠑⢝⣵⣫⢆⢸⠁⠀⢠⣿⢹⣿⢷⣿⣧⡀⢼⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣶⣻⣿⡃⣠⣿⣵⠟⠀⠀⠀⣀⢼⡺⠈⠀⣜⠂⠀⠀⣀⡿⢸⣗⣼⣿⣣⣷⠟⢋⣿⠋⠀⣀⡀⠀
⠀⠀⠀⠀⠀⣠⣿⡿⠋⠈⣷⣿⠃⢐⣯⠀⠈⠈⢸⢀⢮⡲⢹⡪⠘⡵⡫⠀⠐⣌⢠⣾⠏⠙⣿⡘⢧⣹⣮⡟⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣩⣿⣟⣡⣷⡟⠋⠀⠀⠀⣠⠲⡱⠫⣖⠝⡸⠂⣠⢴⠝⠁⠀⢸⣿⣿⡿⠋⠀⢡⣿⣣⡾⡟⠋⠁⠀
⠀⠀⠀⠀⠈⠛⠉⠀⠀⣠⣿⡇⠀⢨⣿⠀⠀⠀⡽⣸⠝⡆⡼⣙⢆⢺⢥⠀⠀⢘⢾⡿⠀⠀⣿⣆⠘⣿⣽⣷⣞⢿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣴⣿⣿⣾⣿⣋⣀⣀⣤⣴⡙⠀⡔⡡⡚⢕⡼⡕⣯⠓⠁⠀⠀⢀⣾⡿⣿⠂⢠⣠⣿⠟⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣿⣄⣿⣿⣄⠀⠴⠝⡞⠀⣝⠎⠈⢳⣝⢝⢥⠀⠀⡯⠁⠀⢐⣿⢻⣼⣿⣿⡏⢻⣿⣝⣧⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⣿⣿⣿⠟⣻⠟⠙⠙⣙⣿⠛⠉⠁⠀⢘⡰⣝⢶⠝⣔⠞⠀⠀⠀⠀⣠⣾⡏⠀⣻⣵⡿⠛⡇⢀⣀⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣼⠟⣽⠇⢽⡿⢳⣿⢹⡇⠀⠀⠀⢠⢷⠀⢀⣞⢞⢦⡹⡄⣀⢯⢄⠀⢨⡿⠸⣿⡏⢽⣇⢨⡿⣿⣿⣧⣤⣤⣽⣦⣶⣶⠶⠶⠶⣶⣾⣵⣶⣿⣯⣿⣿⣿⣿⣶⣶⣾⡋⠀⠀⠀⡠⣔⢗⡽⠊⢁⢼⡪⡂⢀⡠⡤⣰⡟⢹⣃⣴⣿⣧⡶⠾⠟⣿⣿⠿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠻⠋⢠⣿⣴⢿⡇⢰⡿⢿⣿⠀⠀⡠⡏⢱⢁⢴⢃⢟⠄⠙⡽⡨⡓⠫⣧⠐⠋⠈⣿⠀⠸⣿⣼⠛⣿⡟⣿⣄⣀⡀⠉⣹⣿⣿⣿⡻⣿⣿⣿⠟⣫⡿⠻⠳⣦⣤⣴⣫⣄⠀⠀⣀⡮⣪⢮⠫⠤⣔⠼⢵⢔⢽⢅⡅⠀⠀⢀⣼⣿⡿⠉⠹⣄⣰⣾⠟⢁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣾⠃⣼⡏⢸⣿⠀⠀⠀⠀⡸⡮⠃⠀⢝⡕⠑⠭⡯⠂⠀⠘⢳⡀⠀⠉⠀⠀⢿⣟⠀⣸⣷⣸⣿⣿⣟⣻⠿⡷⡶⣿⣿⣷⣿⣿⢼⠿⠷⠟⠛⠛⣉⠽⠻⢙⡨⠏⠃⡹⣪⢂⠴⠕⡩⠎⠉⢂⡡⡕⠆⣠⣶⡿⠋⣾⠁⢀⣠⣾⡿⠿⠟⠛⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⡿⣳⡿⠁⣼⣿⣇⡀⠠⣀⠋⠂⠀⠀⡯⣪⠀⢠⢟⡵⢢⡠⠬⢾⢤⣠⣀⠠⠘⣿⡉⠛⠛⠻⣿⢽⣿⣿⣿⣿⣷⣥⣴⣡⣩⣻⣷⣤⣄⣄⠀⠊⠈⠘⢉⢁⣠⡠⡖⠝⡎⠁⡡⣎⡤⠖⠝⠁⠁⢀⣴⣿⢯⣾⣶⣷⡾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣹⣿⠻⣦⢹⣆⠀⠀⢨⠇⢘⠆⡮⠙⣞⣆⠈⠳⣸⢝⣆⠑⠫⢦⣉⠀⠀⠀⢈⡟⢤⡉⠻⣷⣌⠛⠿⡯⠉⠉⠙⠭⣩⣙⢛⠒⠚⠊⡉⠑⣉⣀⢤⢦⢔⡞⡎⠉⢀⠀⠀⠀⠀⠀⠀⣼⣯⣾⣟⣻⣅⣼⣇⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣿⠃⢹⣯⣀⠙⣷⣿⡄⠀⠞⠀⠨⡯⠃⠀⢱⡣⡅⡓⠤⢯⡺⣆⠀⠀⠱⣳⢬⠉⠓⠣⢆⡙⢦⣈⠁⠀⠒⠊⠖⠦⣄⡠⣠⠴⢝⢽⢝⠝⠞⠝⠚⠎⠯⢚⣗⠽⠚⠎⠏⠗⠅⠀⣀⣤⣶⠿⡿⠋⠉⣻⣡⣵⠿⠿⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣹⣿⡏⠀⢸⣯⠙⢶⣜⣿⡗⢦⣤⣀⠨⡏⠀⠀⠀⠳⡌⠒⣄⠈⣞⠱⢧⣄⣀⠑⢭⡙⢦⢄⠀⠉⠳⣕⢯⢦⣢⡠⣀⢀⡨⠜⢕⠷⢝⣍⡉⠙⠙⠍⢋⡗⠕⠃⢁⣀⣄⢤⠀⣠⡴⣿⡿⠋⠀⣾⠁⣠⣶⣿⣵⣴⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⣿⣦⡠⣹⣿⣧⣀⢈⠻⣷⡁⠀⠀⠀⠀⠹⡄⠈⠱⣺⡂⠘⢲⠬⣘⠮⣖⢤⢹⠲⣔⠤⠴⡹⢵⢕⡯⡮⡙⠚⡮⠫⠲⠴⠴⢝⢗⢖⢼⡕⠖⠚⠊⠃⠁⠀⢀⢰⣁⣾⣋⣤⣶⣾⣿⣿⡟⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⣿⡂⢰⣿⣿⣏⠙⠛⢿⣿⣄⠀⠠⠲⢦⣽⡂⠀⠘⣎⠀⠈⢳⡀⠙⠢⣻⣪⠢⣈⡫⣢⠀⠈⠑⢫⢍⠙⠝⠖⣖⢖⡒⠑⠓⠑⠑⠓⠙⠏⠖⠀⢠⡶⣿⣿⠿⠟⠛⠉⢙⣿⡿⠿⣧⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣿⡆⣾⠇⢹⣿⡄⠀⠀⢸⣿⣧⣄⠀⠀⠙⢿⣶⣌⠙⡥⠀⠀⠹⣄⠀⠀⠣⣇⡀⠉⠊⠛⢦⢤⢄⣉⢶⢄⠀⠀⠁⠉⠛⠲⠴⠄⠀⢀⣀⣀⣠⣾⣿⣩⣀⣀⣀⣠⣼⣋⣻⣴⠞⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⣿⣿⡿⠀⠀⠸⣿⣷⡔⠙⣿⡻⣿⣦⣄⣀⠈⣿⡻⣧⣈⠀⠀⠀⠀⠀⠀⠀⠘⡮⣂⠀⠀⠀⠉⠫⠊⠈⠋⠖⠆⠀⠀⢠⢴⣶⠿⠟⠟⢿⡿⢿⣭⣭⣉⣭⣽⡿⠛⠛⠻⠻⠿⠾⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢹⣯⢻⣆⢹⣧⠸⣿⣏⡛⢷⣼⣧⠮⢿⣶⣶⣦⣤⡀⠀⠀⠉⠋⠛⢶⣤⣀⢀⣠⣤⣤⣴⣴⣴⣴⣴⣤⣯⣥⣀⣀⡀⢘⣤⡶⠟⠋⠋⣛⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⢻⣿⣿⠀⠘⣿⡿⣦⡈⢻⣷⣦⡹⣧⡈⠙⢿⣾⣾⡞⠻⠷⢶⢮⣿⣿⣼⠋⠈⠈⠉⠉⣛⣻⣿⣾⣿⣿⣿⠿⠻⣿⢶⢶⣿⣿⣏⣉⡉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠈⢿⡿⠂⠀⠘⣿⡌⢿⣮⣿⠉⠛⢿⣿⣗⠾⢿⣮⣿⣷⣄⡀⠀⠈⢉⣻⣿⣶⣤⣴⣟⠋⠁⣀⣀⣉⣛⣿⣦⣄⣀⠀⠀⠈⠉⠙⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠹⣿⡇⠀⠀⠙⣿⣷⣤⣻⡏⠉⡻⢿⣶⣟⣙⣩⣽⠛⣯⡝⠛⠷⣿⣿⣝⡙⠙⠙⠛⠛⠿⣷⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢸⣷⡀⠀⠀⠹⣿⠙⣿⣿⡄⠀⠀⠈⠿⣿⣍⠙⠻⢾⣿⣆⠀⠀⠈⠙⠻⠷⡶⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠉⠀⠀⠀⠀⢻⠇⠈⢻⣿⠀⠀⠀⠀⠈⠻⣷⣤⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡗⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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