
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 $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1$$ ($a>0$, $b>0$) with its right focus at $F(c, 0)$.
(1) If one of the asymptotes of the hyperbola is $y=x$ and $c=2$, find the equation of the hyperbola;
(2) With the origin $O$ as the center and $c$ as the radius, draw a circle. Let the intersection of the circle and the hyperbola in the first quadrant be $A$. Draw the tangent line to the circle at $A$, with a slope of $-\sqrt{3}$. Find the eccentricity of the hyperbola.

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