
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 $C$: $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) has an eccentricity of $\sqrt {3}$, and the distance between the imaginary axis endpoint and the focus is $\sqrt {5}$.
(1) Find the equation of the hyperbola $C$;
(2) Given that the line $x-y+m=0$ intersects the hyperbola $C$ at two distinct points $A$ and $B$, and the midpoint of 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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