There are 4 white balls with the numbers $1, 2, 3, 4$ written on them and 3 black balls with the numbers $4, 5, 6$ written on them. When these 7 balls are randomly arranged in a line, the probability that the balls with the same number do not lie adjacent to each other is $\frac { q } { p }$. Find the value of $p + q$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]