
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 \( C: \frac{x^{2}}{ a^{2}}+\frac{y^{2}}{ b^{2}}=1 (a > b > 0) \) with foci \( F_{1} \) and \( F_{2} \) on the left and right, respectively, and \( A \) as the right vertex. A line \( l \) passing through \( F_{1} \) intersects the ellipse at points \( P \) and \( Q \), such that \(\overrightarrow{A P} \cdot \overrightarrow{A Q}=\frac{1}{2}(a+c)^{2} \). Find the minimum eccentricity of the ellipse \( C \).

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