
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 hyperbola $\Gamma$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$), the right vertex is A, and a line parallel to the x-axis intersects $\Gamma$ at points B and C. Let $\vec{AB} \cdot \vec{AC} = m$. If the eccentricity of $\Gamma$ is $\sqrt{2}$, then the set of possible values for $m$ is _________.

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