
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 $f(x) = 2\sqrt{3}\sin x\cos x + 2\cos^2x - 1$ ($x \in \mathbb{R}$),  
(1) Find the intervals where the function $f(x)$ is monotonically decreasing;  
(2) If $A$ is an acute angle of triangle $\triangle ABC$, and satisfies $f(A) = \frac{2}{3}$, find the value of $\sin 2A$.

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