Secure site Re-login using cookies and WWW::Mechanize

I am trying to write a script to automatically login to a secure site.
I want to use cookies to login to the site without specifying the
username and password. This would be similar to "remembering me" option
that some sites have which I believe just uses cookies.

use strict;
use WWW::Mechanize;
use HTTP::Cookies;

my $outfile = "Result.htm";
my $url = "https://www.google.com/accounts/ServiceLogin?";
my $username = "username";
my $password = "password";
my $cookiefile = 'cookies.txt';
my $cookie_jar = HTTP::Cookies->new(File => $cookiefile, autosave =>
1);
my $mech = WWW::Mechanize->new();
$mech->cookie_jar($cookie_jar);
$mech->agent('Mozilla/4.0');
$mech->get($url);
$mech->form_number(1);
$mech->set_visible( $username, $password ) ;
$mech->click();
$mech->follow_link(text => "Click here if your browser does not
automatically redirect you.", n => 1);

##print out current page
my $output_page = $mech->content();
open(OUTFILE, ">$outfile");
print OUTFILE "$output_page";
close(OUTFILE);