A Lucas Sum

Algebra Level 5

Let \(p = \sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}\).

The sum

\[\sum_{k=2}^{\infty} \frac{\lfloor p^{k} \rceil}{2^{k}}\]

can be expressed as \(\frac{a}{b}\) for \(a,b\) coprime, where \(\lfloor \cdot \rceil\) denotes the nearest integer function. Find \(a+b\).

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