Let $F(0)=0, F(1)=\frac{3}{2}$, and $F(n)=\frac{5}{2} F(n-1)-F(n-2)$ for $n \geq 2$. Determine whether or not $\sum\limits_{n=0}^{\infty} \frac{1}{F\left(2^{n}\right)}$ is a rational number. Please provide the integer value of the sum.

Let's think step by step and output the final answer within \boxed{}.
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