
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
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $\overrightarrow{AB} \cdot \overrightarrow{AC} = 8$, $\angle BAC = \theta$, and $a = 4$.

1. Find the maximum value of $b \cdot c$ and the range of $\theta$.
2. Find the extreme values of the function $f(\theta) = 2 \sqrt{3} \sin^2(\frac{\pi}{4} + \theta) + 2 \cos^2 \theta - \sqrt{3}$.

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