Divisibility Duel with Split Draws

In a two-stage experiment, you first draw 2 numbers without replacement from $\{1, 2, 3, 4, 5, 6\}$, then independently draw 1 number uniformly from $\{1, 2, 3, 4, 5\}$. Let $P$ be the product of all three drawn numbers. Find $P(6 \mid P)$, the probability that the product is divisible by $6$.