
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 $(1, \frac{\sqrt{2}}{2})$, and its focal length is $2$.
(1) Find the standard equation of the ellipse $C$;
(2) Let line $l$ passing through the point $P(-2,0)$ intersect 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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