
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 $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ passes through the point $$(1, \frac{\sqrt{2}}{2})$$, and its focal length is 2.
(1) Find the standard equation of ellipse C;
(2) Suppose a line $l$ passing through point P$(-2, 0)$ intersects ellipse C at two distinct points A and B, and point $$G(0, -\frac{1}{2})$$, if $|GA| = |GB|$, find the equation of line $l$.

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