
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 a triangle $\triangle ABC$ with area $S$ that satisfies $2-\sqrt{3} \leq S \leq 1$, and $\overrightarrow{AC} \cdot \overrightarrow{CB} = -2$, $\angle ACB = \theta$.
(1) If $\overrightarrow{m} = (\sin 2A, \cos 2A)$ and $\overrightarrow{n} = (\cos 2B, \sin 2B)$, find the range of values for $|\overrightarrow{m} + 2\overrightarrow{n}|$.
(2) Find the maximum value of the function $f(\theta) = \sin(\theta + \frac{\pi}{4}) - 4\sqrt{3}\sin\theta\cos\theta + \cos(\theta - \frac{\pi}{4}) - 2$.

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