Let \( S=\{(m,n): m,n\in \{1,2,3,\ldots,50\}\} \). If the number of elements \( (m,n) \) is \( S \) such that \( 6^{m}+9^{n} \) is a multiple of \( 5 \) is \( p \) and the number of elements \( (m,n) \) in \( S \) such that \( m+n \) is a square of a prime number is \( q \), then \( p+q \) is equal to _________.
