Something

#!/usr/bin/perl
$filecount=1;
open OUT, ">output." . $filecount;
while (<>) {
  if ($_=~/^$/) {
    $blankcount++;
    if ($blankcount==4) {
      print STDERR "New section found, opening output." . $filecount;
      close OUT;
      $blankcount=0;
      $filecount++;
      open OUT, ">output." . $filecount;
    }
  } else {
   print OUT $_;
  }
}
close OUT;