Balanced strings using two kinds of brackets

Consider strings of length $2n$ built from the four symbols (, ), {, }. Call a string proper if every bracket is closed by a bracket of the same kind and the brackets are correctly nested, so (){} and {()} are proper while {(}) and ((} are not. Let $P_n$ denote the number of proper strings of length $2n$. Find $P_5$.

$1344$. Ignoring the bracket type, a proper string is a balanced parenthesisation, counted by the Catalan number $C_n$; then each of the $n$ matched pairs independently chooses its type. So $P_n=2^nC_n$ and $P_5=32\cdot42=1344$.