
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 $xOy$, point $P\left( \frac {1}{2}, \frac { \sqrt {3}}{2}\right)$ is given. Rotating vector $\overrightarrow{OP}$ around the origin $O$ in the counterclockwise direction by an angle of $x$ radians results in 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 $g(x)=f(x) \cdot f(x+ \frac {\pi}{3})$, find the range of the function $g(x)$.

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