
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 $f(x) = \cos x \cdot \sin\left(\frac{\pi}{6} - x\right)$,  
(1) Find the interval where $f(x)$ is monotonically decreasing;  
(2) In $\triangle ABC$, the sides opposite angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. If $f(C) = -\frac{1}{4}$, $a=2$, and the area of $\triangle ABC$ is $2\sqrt{3}$, find the length of side $c$.

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