Let \(p,q\) be two prime numbers such that \[6\mid pq+(p+1)(q+1)(p+q)\]

If \((p,q)=(p_1,q_1),(p_2,q_2),\ldots, (p_n,q_n)\) are all solutions for the ordered pair \((p,q)\) then find \[\sum_{i=1}^np_i+q_i.\]

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