
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 the function $f(x)=2 \sqrt {3}\sin x\cos x+2\sin ^{2}x$.  
$(1)$ If $f(x)=0$, and $x\in\left(- \frac {\pi}{2},\pi\right)$, find the value of $x$;  
$(2)$ Move the graph of the function $f(x)$ to the left by $\frac {\pi}{3}$ units, and then stretch all the x-coordinates of the points on the graph by $2$ times (the y-coordinates remain unchanged) to obtain the graph of the function $g(x)$. If the curve $y=h(x)$ is symmetric to $y=g(x)$ about the line $x= \frac {\pi}{4}$, find the range of the function $h(x)$ on $\left(- \frac {\pi}{6}, \frac {2\pi}{3}\right]$.

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