\[\large 4^{a}-b^{2}=c\]

There are a finite number of ordered triples \((a,b,c)\) that can satisfy the above equation, where \(a,b,\) and \(c\) are all primes.

Find the sum of all distinct primes that are a part of a solution set for the above equation.

\[\]
**Details and Assumptions:**

If the two solutions were \((4,5,6), (4,6,8)\), then the answer would be \(4+5+6+8=23.\)

\(1\) is \({\color{red} \text{not}}\) a prime number.

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