
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
The right focus of the hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1 \, (a > 0, b > 0)$ is $F(c, 0)$.  
(1) If one of the asymptotes of the hyperbola is $y = x$ and $c = 2$, find the equation of the hyperbola;  
(2) A line $l$ that passes through the origin and has a slope angle of $30^\circ$ intersects the right branch of the hyperbola at point $A$, and $\triangle OAF$ is an isosceles triangle with $AF$ as its base. Find the value of the eccentricity $e$ of the hyperbola.

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