Consider the lines $L_1$ and $L_2$ defined by $L_1: x \sqrt{2}+y-1=0$ and $L_2: x \sqrt{2}-y+1=0$ For a fixed constant $\lambda_{\text {, }}$ let $C$ be the locus of a point $P$ such that the product of the distance of $P$ from $L_1$ and the distance of $P$ from $L_2$ is $\lambda^2$. The line $y=2 x+1$ meets $C$ at two points $R$ and $S$, where the distance between $R$ and S is $\sqrt{270}$. Let the perpendicular bisector of $R S$ meet $C$ at two distinct points $R^{\prime}$ and $S^{\prime}$. Let $D$ be the square of the distance between $\mathrm{R}^{\prime}$ and $\mathrm{S}^{\prime}$.
The value of $\lambda^2$ is $\_\_\_\_$ .