3 Consecutive Cards II

$k \geq 1$ standard decks are combined together to form a large deck. $3$ cards are selected uniformly at random from this large deck without replacement. Let $p_k$ be the probability these three cards form a three card straight (following similar rules to the standard poker straight, which means that an Ace can either be the lowest card or the highest card) when there are $k$ decks combined. Find $\displaystyle \lim_{k \rightarrow \infty} p_k$. Note that a straight flush is allowed in this.