
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{3}}{2}$ and points A($a$, 0), B(0, $b$), O(0, 0), where the area of triangle $\triangle OAB$ is 1,
1. Find the equation of ellipse $C$.
2. Find the equation of the line with slope 2 that intersects the ellipse at two points $P$ and $Q$ such that $OP \perp OQ$.
3. Determine whether there exists a point $E$ on the x-axis such that for any line passing through $E$ and intersecting the ellipse at two points $M$ and $N$, the sum $\frac{1}{|EM|^2} + \frac{1}{|EN|^2}$ is a constant value. If such a point exists, find the coordinates of $E$ and the corresponding constant value.

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