
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 line $l\_1$ intersects with the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ at points $A$ and $B$, and the x-coordinate of the midpoint $M$ of $AB$ is $b$. The line $l\_2$ passes through $M$, is perpendicular to line $l\_1$, and also passes through the right focus of the hyperbola $C$. Calculate the eccentricity of the hyperbola $C$.

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