Take row pairs in one pandas dataframe and iterate through another dataframe

Question

I would like to iterate through row pairs of df_a, comparing firstname1 to lastname1 and firstname2 to lastname2. So for each row pair (Ex. firstname1 & lastname1), I would then like to iterate through df_b and identify non-overlapping coordinate ranges (coordinate_start, coordinate_end) for those names, for example the coordinate ranges assigned for firstname1 do not overlap with any of the coordinate ranges for lastname1.

Starting with df_a & df_b:

a = {'ID_a': ['firstname1', 'firstname2'], 'ID_b': ['lastname1', 'lastname2']}

    
df_a = pd.DataFrame(a)



b = {'coordinate_start' :[1,6,20,35,51,1,7,15,40,51,70,85,91,70,80,94], 'coordinate_end':[5,15,27,50,55,5,14,19,47,55,78,90,93,78,84,100],
 'name': ['firstname1', 'firstname1','firstname1', 'firstname1','firstname1',
 'lastname1','lastname1','lastname1','lastname1','lastname1',
 'firstname2','firstname2','firstname2',
 'lastname2','lastname2', 'lastname2'
 ]}

df_b = pd.DataFrame(b)

I would like to return df_c, which contains the non-overlapping coordinates and the name it's associated with:

c = {'unique_coordinate_start': [20,85,80,91,94],
'unique_coordinate_end': [27,90,84,93,100],
'name': ['firstname1','firstname2', 'lastname2','firstname2','lastname2']}

df_c = pd.DataFrame(c)