
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 $$C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ (where $a > 0, b > 0$) with eccentricity $\sqrt{3}$, the equation of its right directrix is $x = \frac{\sqrt{3}}{3}$.  
(I) Find the equation of the hyperbola $C$;  
(II) A straight 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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