
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
Let $\triangle ABC$ be an inscribed triangle of the ellipse $\Gamma: \frac{x^{2}}{4} + y^{2} = 1$, where $A$ is the intersection of the ellipse $\Gamma$ with the positive x-axis. The product of the slopes of the lines $AB$ and $AC$ is $-\frac{1}{4}$. Let $G$ be the centroid of $\triangle ABC$. Find the range of values of $|GA| + |GB| + |GC|$.

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