Infinite Multiplication \(=\) Infinite Addition

Algebra Level 5

\[\small \sqrt{5\sqrt{5\sqrt{5\sqrt{5\sqrt{\cdots}}}}} = \sqrt{x^2 + \sqrt{4x^2 + \sqrt{16x^2 + \sqrt{64x^2 + \sqrt{\cdots}}}}}\]

If \(x\) is a positive number satisfying the equation above, find \(x\).

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