
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 an ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\left(a > b > 0\right)$ with its right focus at $F(1,0)$, and its symmetric point $Q$ with respect to the line $y=bx$ lies on the ellipse. Find the eccentricity $e=$____ and the area of $\triangle FOQ$ is $S_{\triangle FOQ}=$____.

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