
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 \( E: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \), points \( A(0,2) \), and \( P(-3,2) \). A line \( l \) passing through point \( P \) intersects the ellipse \( E \) at two distinct points \( B \) and \( C \). The lines \( AB \) and \( AC \) intersect the x-axis at points \( M \) and \( N \), respectively.
1. If \( |MN| = 6 \), find the equation of the line \( l \).
2. Find the equation of the locus of the circumcenter of the triangle \( PMN \).

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