Create genome-unique intervals from overlapping records in a bed file

Let there exist a bed file, a, with 3 overlapping records:

a

chr1      1      20      s1      1      +      
chr1      5      20      s2      1      +      
chr1      10    20      s3      1      +

I want to write a function which would parse a and return a bed-file, bm containing genome-unique windows along with a list of bed-records which span those windows, i.e.

b

chr1      1      5      s1      1      +      
chr1      5      10      s1,s2,s3      1      +      
chr1      10      20      s1, s2, s3      1      +      

The goal of the function is to produce genome-unique bed records (essentially an index), from which combinations of records can be used to produce each of the original bed records.

I wouldn’t know where to start with something like this. Any suggestions are greatly appreciated.