
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 vectors $\overrightarrow{a}=(\cos \frac{3x}{2}, \sin \frac{3x}{2})$ and $\overrightarrow{b}=(\cos \frac{x}{2}, \sin \frac{x}{2})$.
- (1) Knowing that $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$ and $x \in [0, \frac{\pi}{2}]$, find $x$.
- (2) Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$. Determine the intervals where $f(x)$ is monotonically decreasing.

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