Prime Time

Let \(P\) and \(Q\) be positive primes such that \[\large 2^{P - 1} + 5PQ = Q^2 + 4P^2 + 7\] If the ordered pairs satisfying the equation are \((P_1, Q_1)\), \((P_2, Q_2)\), \((P_3, Q_3)\), \(\cdots\) \((P_n, Q_n)\) for some positive integer \(n\), find the value of \(\displaystyle \sum_{k = 1}^{n} (P_k + Q_k)\).

This problem is part of the set "Symphony"

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