
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 a hyperbola $C$ with equation $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) has eccentricity $\sqrt {3}$, and the distance from its foci to the asymptotes is 2.  
(1) Find the equation of the hyperbola $C$.  
(2) It is known that the line $x - y + m = 0$ intersects the hyperbola $C$ at two distinct points $A$ and $B$, and the midpoint of the line segment $AB$ lies on the circle $x^{2} + y^{2} = 5$. Find the value of $m$.

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