Rerolling three dice to get the same multiset

Roll three fair six-sided dice and note the values shown (ignoring which die shows which). Then roll all three again. What is the probability that the second roll shows exactly the same collection of values as the first?

$83/3888$. Condition on the first roll's pattern: all equal (prob $\tfrac1{36}$, then $1$ matching ordered outcome), one pair (prob $\tfrac5{12}$, $3$ matching outcomes) or all distinct (prob $\tfrac59$, $6$ matching outcomes). Total: $\tfrac{1}{216}\left(\tfrac1{36}+\tfrac{15}{12}+\tfrac{30}{9}\right)=\tfrac{83}{3888}$.