
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 the function $f(x) = \frac{(1+\cos 2x)^2 - 2\cos 2x - 1}{\sin\left( \frac{\pi}{4} + x \right)\sin\left( \frac{\pi}{4} - x \right)}$.  
(1) Find the value of $f\left(-\frac{11\pi}{12}\right)$.  
(2) For $x \in \left[0, \frac{\pi}{4}\right)$, find the maximum and minimum values of $g(x) = \frac{1}{2}f(x) + \sin 2x$.

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