9、 (16 points) As shown in the figure, there are $2^{n}$ points on the parabola $y^{2}=2 p x(p>0)$, denoted as $P_{1}, P_{2}, \ldots, P_{2^{n}}$, and $F$ is the focus of the parabola. The angles between the lines connecting each adjacent pair of points and $F$ are equal, i.e.,
$$
\angle P_{1} F P_{2}=\angle P_{2} F P_{3}=\cdots=\angle P_{2^{n}} F P_{1}
$$

Let the distance from $P_{i}$ to the directrix be $d_{i}$. Find
$$
\sum_{i=1}^{2^{n}} \frac{1}{d_{i}}
$$