A long way to do it with a loop and a flag

#!/bin/perl5

use strict;
use warnings;

my $dollars;
while (<DATA>){
  chomp;
  if ($dollars){
    s/^line(\d+)/name$1/;
    $dollars--
  }
  print "$_\n";
  $dollars++ if /\${4}/
}

__DATA__
#something
$$$$
line1
#something
$$$$
line2
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