I calculate 18 visits for $459
$099 Annual fee
$000 3 free visits
$360 15 paid visits @ $24 each
$459 18 total visits
...but I don't think the Citi Platinum American Express fits into a two card solution unless gj83 chooses to give up on airline miles entirely in favor of Thank You points.
I tend to exclude one-time bonuses from calcs. You could also exclude the first year annual fee, since it is waived.