Do you already have pre-determined file names in place?

my @files = qw( A.csv B.csv C.csv D.csv E.csv );
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I guess you might want to come up with something extensible so that the list can automatically expand.

The $count variable is really unnecessary here. You could use $. (unless there are criteria where you might skip lines)

You probably don't want to open the OUTPUT every line. So... (psuedo-untested-code to give you an idea)

 open( OUTPUT, $files[0] ) or die... 
 while( <DISTLISTS> ) {
  if (not $. % 200) {
   close OUTPUT;
   open OUTPUT, $files[$./200] or die...
  }
  print OUTPUT, $dist_split;
 }
 close OUTPUT;
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Just an idea.