Is it really Inequality Problem-2?

Algebra Level 5

Let \(p,q,a\) be real numbers such that \(p^{2}+q^{2}-2p=0\) , then the minimum value of \[\sqrt{p^{2}+q^{2}+2a^{2}+16-2ap-8a-8q+2aq}\] is \[\frac{m}{\sqrt{e}}-d\] where \(m,e\) are prime numbers and \(d\) is a positive integer . Find \(m+e+d\).

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