
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 $\overrightarrow{a}=(\cos x, \frac{1}{2})$ and $\overrightarrow{b}=( \sqrt {3}\sin x,\cos 2x)$, where $x\in R$, let the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}$.
(1) Find the smallest positive period of $f(x)$;
(2) Find the interval(s) where the function $f(x)$ is monotonically decreasing;
(3) Find the maximum and minimum values of $f(x)$ on $[0, \frac{\pi}{2}]$.

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