Here is a more newbie-friendly version of jeffa's code. Consider it a learning tool.

my $file_num = 1;
while (<IN>) {
   if (/^(\d+)\s*$/) {
      my $count = $1;
      my $fname = sprintf("new_file%02d.txt", $file_num);
      open(OUT,">$fname") or die "Cannot open file $fname: $!\n";
      for(1..$count){
         my $rec = <IN>; 
         print OUT $rec;
      }
      close OUT; 
      $file_num++;
   } else {
     # ignore the file separator =======
   }
}
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