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Calculus Level 5

Consider all pairs of real values \( (p,q) \) that satisfy

\[ p^{2} + q^{2} +13 = 8p + 2q. \]

The maximum and minimum value of

\[\sqrt{p^{2} + q^{2} -2p + 6q +9}\] are \(M \) and \(m\) respectively. Find the value of \(4(\frac{M}{m})^{2}\).

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