Three-Card Straight From Combined Decks

You take $k \geq 1$ standard 52-card decks and shuffle them together into one big deck. You draw 3 cards uniformly at random without replacement. A "three-card straight" means the three cards have three consecutive ranks (e.g., 5-6-7 or Q-K-A). An Ace can serve as either the low card (A-2-3) or the high card (Q-K-A). Suits don't matter -- straight flushes count too. Let $p_k$ be the probability the three drawn cards form a three-card straight. Find $\displaystyle\lim_{k \to \infty} p_k$.