
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 $F_1$ and $F_2$ are the left and right foci of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, (a > b > 0)$, respectively, and $O$ is the origin of the coordinate system. The focal distance of ellipse $C$ is 6. A chord AB through $F_1$ with endpoints $A$ and $B$, and the perimeter of $\triangle ABF_2$ is $12\sqrt{2}$.

(1) Find the standard equation of ellipse $C$;  
(2) Given points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are two different points on ellipse $C$, and the midpoint of segment $PQ$ is $M(2, 1)$. Find the equation of line $PQ$.

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