
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
Let $P$ be a moving point on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$. A tangent line at point $P$ to the ellipse intersects the circle $\odot O : x^{2}+y^{2}=12$ at two points $M$ and $N$. The tangents to the circle $\odot O$ at points $M$ and $N$ intersect at point $Q$.

(1) Find the equation of the locus of point $Q$.
(2) If $P$ is in the first quadrant, find the maximum area of triangle $\triangle OPQ$.

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