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Combing degrees and minutes data columns, accounting for rolling over to next degree. (Python)

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I have a have a weird format for some data that I'm stuck working with. it's a csv file that includes a timestamp column, a latitude minutes column which is a float, and longitude minutes column (float). The data does include a starting Latitude and Longitude in degrees as well, but only a starting value. I need to combine the latitude and longitude minutes columns with the starting values and convert to degrees and decimal degrees. I'm stuck with this format.

lat = 45long = 75data = {'time': ['2024-03-27 12:00:00', '2024-03-27 12:00:01', '2024- 03-27 12:00:02', '2024-03-27 12:00:03', '2024-03-27 12:00:04', '2024-03-27 12:00:05'    ],'lat_minutes': [        59.9955, 59.9963, 0.0180, 0.0230, 0.0050, 59.6500    ],'long_minutes': [        59.0250, 59.0750, 0.0020, 0.1850, 0.0750, 59.075    ]}df = pd.DataFrame(data)    time                 lat_minutes       long_minutes0   2024-03-27 12:00:00  59.9955            59.0251   2024-03-27 12:00:01  59.9963            59.0752   2024-03-27 12:00:02  0.0180             0.0023   2024-03-27 12:00:03  0.0230             0.1854   2024-03-27 12:00:04  0.0050             0.0755   2024-03-27 12:00:05  59.6500            59.075

I need the output to make a new column with degrees.decimal format based on adding the minutes column to starting lat and long values accounting for adding a whole degree once teh minutes roll past 60, and subtracting a degree if they are decreasing and pass o going to 59.9999.

output should be this:

    time                 lat_minutes  long_minutes   Latitude  Longitude0   2024-03-27 12:00:00  59.9955         59.025     45.999925   75.98375      1   2024-03-27 12:00:01  59.9963         59.075     45.9999383  75.9845832   2024-03-27 12:00:02  0.0180          0.002      46.00030    76.00033   2024-03-27 12:00:03  0.0230          0.185      46.00383    76.00308 4   2024-03-27 12:00:04  0.0050          0.075      46.00125    76.001255   2024-03-27 12:00:05  59.6500         59.075     45.99416    75.98458

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