Feeling left out, as everyone else around here seems to be running around with various polychromatic items attached to their foreheads, you and I decided to play a friendly variant of Indian Poker. Because I'm partial to pointy things that resemble weapons, I take all the spades (13 cards) from a standard 52 card deck, shuffle them, and deal us each one. We each then affix the card to our distinctly-bare-up-until-now foreheads without looking at our own cards. Then we take turns declaring "higher" (if we think our card is higher than the other's), "lower" (if we think our card is lower than the other's), or we say "pass" (if we don't know). Since you and I both hate to be wrong, neither will declare unless we have a logical reason to do so.
What is the maximum number of iterations it will take before one us can declare?