There is method to my madness

Find the sum of all \(p,q,s\) such that

\(2^m.p^2+1=q^s\)

and \(p,q,s\) are odd primes and \(m\) is not divisible by 4.

Clarification:

For every such triple, add the values of \(p,q\) and \(s\) to obtain the answer.

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