Looking at the WWW::Mechanize and LWP::UserAgent docs, looks like this (or these methods at least) should get you in the right direction:

use WWW::Mechanize;
my $mech = WWW::Mechanize->new();

$mech->get( $url1 );

$mech->add_header( 'Cookie' => $cookie1 );
$mech->get( $url1 );

$mech->delete_header( 'Cookie' );
$mech->add_header( 'Cookie' => $session );
$mech->default_headers->authorization_basic($browser_username,$browser
+_password);
$mech->get( $url3 );

...
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