
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given that the left focus of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $F$, the left and right vertices are $A$ and $B$ respectively. A line $l$ passing through point $F$ with a slope of $\frac{\pi}{4}$ intersects the ellipse at points $C$ and $D$. The eccentricity of the ellipse $C$ is $\frac{\sqrt{3}}{2}$, and $\overset{→}{AC} \cdot \overset{→}{AD} - \overset{→}{BC} \cdot \overset{→}{BD} = -\frac{32\sqrt{3}}{5}$.

1. Find the equation of the ellipse $C$.
2. If $P_1$ and $P_2$ are two different points on the ellipse, perpendicular to the $x$-axis, and circle $E$ passes through points $P_1$ and $P_2$. If no point on the ellipse is inside circle $E$, then circle $E$ is called an inscribed circle of the ellipse. Does ellipse $C$ have an inscribed circle passing through point $F$? If yes, find the coordinates of point $E$; if no, explain the reason.

Present the answer in LaTex format: \boxed{Your answer}
user