
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
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad (a>0, b>0),
$$

with left and right foci $F_{1}$ and $F_{2}$ respectively. A perpendicular is drawn from $F_{2}$ to the $x$-axis, intersecting the hyperbola at points $A$ and $B$. The radius of the inscribed circle of $\triangle A B F_{1}$ is $a$. Determine the eccentricity of this hyperbola.

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