Dealing until two kings or a king and an ace

A well-shuffled $52$-card deck is turned over one card at a time. Dealing stops the moment either a second king has appeared, or at least one king and at least one ace have both appeared, whichever comes first. What is the expected number of cards turned over?

$\tfrac{1219}{90}\approx13.54$. Dealing always ends at the second card from the set {kings, aces} unless the first such card is an ace, in which case it ends at the first king. By the equal-gaps argument the second of the 8 special cards sits at position $\tfrac{106}{9}$ on average (independent of labels), while given an ace comes first the first king averages position $\tfrac{689}{45}$. Each case has probability $\tfrac12$, so the mean is $\tfrac12\big(\tfrac{106}{9}+\tfrac{689}{45}\big)=\tfrac{1219}{90}$, confirmed by exact recursion.