
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 $\overrightarrow{m} = (2\sin x, -\sqrt{3})$ and $\overrightarrow{n} = (\cos x, 2\cos^2 x - 1)$, and a function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n} + 1$.

(1) Find the smallest positive period of $f(x)$;
(2) Find the range of $f(x)$ on the interval $[\frac{\pi}{4}, \frac{\pi}{2}]$.

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