
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 that the center of circle $M$ is $M$ with equation $(x+\sqrt{3})^2 + y^2 = \frac{27}{2}$, and the center of circle $N$ is $N$ with equation $(x-\sqrt{3})^2 + y^2 = \frac{3}{2}$. A moving circle is tangent to circle $M$ internally and tangent to circle $N$ externally.

1. Find the equation of the locus of the center $C$ of the moving circle.
2. A line $l$ passing through point $P(2,0)$ intersects the curve $C$ at points $A$ and $B$. Point $Q$ is any point on the line $x=3$. The slopes of lines $QA$, $QP$, and $QB$ are denoted as $k_{1}$, $k_{2}$, and $k_{3}$, respectively. Find the relationship between $k_{1}$, $k_{2}$, and $k_{3}$, and provide a proof.

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