\[\large{ 2^{a+1} - 2^a + 2 = \ 2^b - 2^c }\]

Evaluate the sum of the sums of all ordered tuples of integers \((a,b,c)\) that exist as solutions to the equation above.

**Details and Assumptions:**

\(a, b,\) and \(c\) are neither necessarily distinct nor positive.

The "sum of a tuple" is the sum of its members.

Bonus: For extra fun, consider the same question but with the equation \(2^{a+1}-2^a+1=2^b-2^c\).

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