GAIC Math 2024 Problem 110

In the Cartesian coordinate system xOy, let $\Gamma_{1}$ be the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and $\Gamma_{2}$ be the parabola $y^{2}=\frac{1}{2}ax$. They intersect at points A and B, and P is the rightmost point of $\Gamma_{1}$. If points O, A, P, and B are concyclic, then the eccentricity of $\Gamma_{1}$ is_____.