Factorization is an art (3)

Algebra Level 4

\[\large(64^{x}-128)^{3} + (128^{x}-64)^{3} = (128^{x}+64^{x}-192)^{3}\]

If the sum of all real root(s) of the above equation is \(\dfrac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, and \(a+b = c^2\), find \(|c|\).

Notation: \( | \cdot | \) denotes the absolute value function.

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