
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. The eccentricity $e = \frac{1}{2}$. $P$ is a moving point on the ellipse, and the maximum area of $\triangle PF\_1F\_2$ is $4\sqrt{3}$.

(1) Find the equation of the ellipse;

(2) If $A$, $B$, $C$, $D$ are four distinct points on the ellipse, $AC$ and $BD$ intersect at point $F\_1$, and $\overrightarrow{AC} \cdot \overrightarrow{BD} = 0$, find the range of values for $|\overrightarrow{AC}| + |\overrightarrow{BD}|$.

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