Divisible by 6

Number Theory Level 4

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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