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

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.

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**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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