A Ratio from a Diameter Circle

Let $M,N$ be two points on the ellipse $\dfrac{x^2}4+y^2=1$ with the line $MN$ of slope $1$. Let $P$ be the reflection of $M$ across the $x$-axis. The circle with diameter $PN$ meets the $x$-axis at points $D$ and $E$, with $|OD|\geqslant|OE|$ ($O$ the origin). Find $\dfrac{|OD|}{|OE|}$.