
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 its two foci being $F_1$ and $F_2$, and $|F_1F_2| = 2\sqrt{3}$, the length of the major axis is 4, and P is any point on ellipse C.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) Find the range of values for $\overrightarrow{PF_{1}} \cdot \overrightarrow{PF_{2}}$;  
(Ⅲ) Suppose the left and right vertices of the ellipse are A and B, respectively, line PA intersects line $l: x=4$ at point M, connect MB, and the line MB intersects ellipse C at another point Q. Determine whether line PQ passes through a fixed point, and if so, find the coordinates of the fixed point; if not, explain why.

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