
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
In the Cartesian coordinate system $xoy$, given the point $P(\frac{1}{2}, \frac{\sqrt{3}}{2})$, the vector $\overrightarrow{OP}$ is rotated counterclockwise around the origin $O$ by $x$ radians to obtain the vector $\overrightarrow{OQ}$.

(1) If $x=\frac{\pi}{4}$, find the coordinates of point $Q$.

(2) Given the function $f(x)=\overrightarrow{OP} \cdot \overrightarrow{OQ}$, let $g(x)=f\left(x\right) \cdot f\left(x+ \frac{\pi}{3}\right)$, find the range of the function $g(x)$.

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