
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
In the Cartesian coordinate plane $(xOy)$, an ellipse with the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance from its right focus $F$ to the line $l:x=-\frac{a^2}{c}$ is 3.
(1) Find the equation of the ellipse;
(2) A line passing through $F$ intersects the ellipse at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects line $l$ and $AB$ at points $P$ and $C$, respectively. If $|PC|=2|AB|$, find the equation of line $AB$.

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