
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 $\vec{a} =( \cos x, -\frac{1}{2} )$ and $\vec{b} =(\sin x, \cos 2x)$, where $x\in \mathbb{R}$. Define the function $f(x) = \vec{a} \cdot \vec{b}$.

(I) Find the smallest positive period of $f(x)$.

(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[0, \frac{\pi}{2} \right]$.

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