
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 E: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (where $a > b > 0$) with its left and right foci denoted as $F_1$ and $F_2$ respectively, the eccentricity $e = \frac{1}{2}$, and the focal distance is 4.
(I) Find the equation of ellipse E.
(II) If lines from $F_1$ to $A$, $B$, $C$, $D$ on the ellipse E are such that $\overrightarrow{F_1A}$ is parallel to $\overrightarrow{F_1C}$, $\overrightarrow{F_1B}$ is parallel to $\overrightarrow{F_1D}$, and $\overrightarrow{AC} \cdot \overrightarrow{BD} = 0$, find the minimum value of $|\overrightarrow{AC}| + |\overrightarrow{BD}|$.

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