A line $L$ has a fixed slope $m=2$ and passes through a variable point $(t, t^2)$ on the parabola $y=x^2$. As $t$ varies, the line $L$ intersects the $y$-axis at $P(0, y_p)$. Find the locus of the midpoint of the segment connecting $(0, 0)$ and $P(0, y_p)$ as $t$ changes.