
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
Consider the hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively, and eccentricity $e$. A line passing through $F\_2$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F\_1AB$ is an isosceles right triangle with $A$ as the right-angled vertex, then find the value of $e^2$.

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