Coordinate Geometry: JEE Main 2025 Mathematics question

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\tfrac{x^2}{a^2}+\tfrac{y^2}{25}=1$ and the hyperbola $\tfrac{x^2}{16}-\tfrac{y^2}{b^2}=1$, respectively. If $b<5$ and $e_1e_2=1$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is: