How can it be true?

Algebra Level 3

\[{2^x + 2^{x+1} + 2^{x+2} + \ldots + 2^{x+2015} = 4^x + 4^{x+1} + 4^{x+2} + \ldots + 4^{x+2015}}\]

If \(x\) satisfies the equation above and it can be represented as

\[\large{\log_D \left( \dfrac{A}{1+B^C} \right)}\]

for positive integers \(A,B,C,D\) where \(B\) is prime, then determine the smallest value of \(A + B + C +D\).

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