
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{2}=1 (a > \sqrt{2})$ with left and right foci $F_{1}$ and $F_{2}$, respectively, eccentricity $e$, and line $l$: $y = ex + a$. Let $P$ be the point symmetric to point $F_{1}$ with respect to line $l$. If $\triangle PF_{1}F_{2}$ is an isosceles triangle, find the value of $a$.

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