
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 that \( F_{1} \) and \( F_{2} \) are the foci of the ellipse \( \Gamma: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with \( a > b > 0 \), \( P \) is a point on \( \Gamma \), and the perimeter of \( \triangle PF_{1}F_{2} \) is 6, and that the minimum value of \( \frac{4}{a} + \frac{1}{c} \) is 3, a line passing through the point \( Q(-4, 0) \) intersects \( \Gamma \) at two distinct points \( A \) and \( B \). Determine the range of values for \( |QA| \cdot |QB| \).

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