
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 $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ ($a > 0$, $b > 0$) with a focal length of 4 and an eccentricity of $\frac{2\sqrt{3}}{3}$,
(1) Find the standard equation of hyperbola C;
(2) If the line $l: y = kx + m$ ($k \neq 0$, $m \neq 0$) intersects the hyperbola C at two distinct points, C and D, and both C and D lie on the same circle centered at the point A (0, -1), determine the range of the real number m.

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