
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with left and right foci $F_1$ and $F_2$, respectively, and an eccentricity of $\frac{1}{2}$. $P$ is a moving point on the ellipse (different from the left and right vertices). If the maximum area of $\triangle F_1F_2$ is $\sqrt{3}$:
1. Find the equation of ellipse C;
2. If a line $l$ passes through point $F_1$ and intersects ellipse C at points A and B, is there a point Q on the x-axis such that $\overrightarrow{QA}\cdot \overrightarrow{QB}$ is a constant value? If so, find the coordinates of point Q; if not, explain the reason.

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