
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 an area of 2, and satisfying $0 < \vec{AB} \cdot \vec{AC} \leq 4$, let the angle between $\vec{AB}$ and $\vec{AC}$ be $\theta$.  
(1) Determine the range of possible values for $\tan\theta$.  
(2) Find the maximum and minimum values of the function $f(\theta) = 2\sin^2\left(\frac{\pi}{4} + \theta\right) - \sqrt{3}\cos(2\theta)$.

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