
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 two foci $F_{1}$, $F_{2}$, ($|F_{1}F_{2}| = 2\sqrt{3}$), and the major axis length is $4$. $P$ is any point on the ellipse $(C)$.

(I) Find the equation of the 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 other intersection point of line $MB$ with ellipse $(C)$ is $Q$. Determine whether line $PQ$ passes through a fixed point, and if so, find its coordinates; if not, explain why.

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