
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 \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), let \(A\) be the left vertex, \(F(c, 0)\) be the right focus, and \(2b\), \(a\), and \(c\) form a geometric progression. A line through the point \(F\) intersects the ellipse at points \(M\) and \(N\). The lines \(AM\) and \(AN\) intersect the right directrix at points \(P\) and \(Q\) respectively. Determine the size of the angle \(\angle PFQ\).

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