\[\Large 1+\frac{2}{1+\frac{2}{ 1+\frac{2}{\ddots+\frac{2}{x}}}}=2\]

The above infinite continued fraction can be viewed as a recurrence relation: \(a_0=x\), \(a_{k+1}=1+\frac{2}{a_{k}}\) for \(k\ge 0\) and \(\displaystyle\lim_{k \to \infty}a_k=2\).

What is the range of integers \(x\) that satisfy this condition?

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