
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 an ellipse (C): $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(1, \frac{\sqrt{3}}{2})$, with the left focus $F_{1}(-\sqrt{3}, 0)$.
(1) Find the equation of the ellipse (C);
(2) A circle $x^{2} + y^{2} = 1$ has a tangent line (l) passing through the point $(m, 0)$, intersecting the ellipse (C) at points A and B. Represent $|AB|$ as a function of $m$, and find the maximum value of $|AB|$.

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