
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) with eccentricity $e = \frac{2\sqrt{3}}{3}$, and the distance from the origin to the line passing through points A(0, -b) and B(a, 0) is $\frac{\sqrt{3}}{2}$,
(1) find the equation of the hyperbola;
(2) line $y = kx + m$ ($k \neq 0, m \neq 0$) intersects the hyperbola at two distinct points C and D, and both points C and D lie on the same circle centered at point A. Find the range of values for $m$.

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