Collision Pattern in Two-Round Sampling

In round 1, you draw 3 numbers independently and uniformly at random (with replacement) from $\{1, 2, 3, 4\}$. In round 2, you draw 2 numbers independently and uniformly at random (with replacement) from $\{3, 4, 5, 6\}$. The two pools overlap at $\{3, 4\}$. Across all 5 draws, find $P(\text{exactly one value appears more than once, and that value appears exactly twice})$. Express your answer as a reduced fraction.