
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
Suppose \( F \) is a focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4^{2}}=1\) (where \(a > 0\) and \(a \neq 2\)). Let \(AB\) be a segment of length 3 with endpoints that move on the ellipse. Then, when \(|AF| \cdot |BF|\) reaches its maximum value, \(a = \)

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