
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 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 at $F_1(-\sqrt {3}, 0)$.
(1) Find the equation of ellipse C;
(2) A tangent line $l$ to the circle $x^2 + y^2 = 1$ passing through the point $(m, 0)$ intersects ellipse C at points A and B. Express $|AB|$ as a function of $m$, and find the maximum value of $|AB|$.

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