
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 points $A(x_{1},f(x_{1}))$ and $B(x_{2},f(x_{2}))$ are any two points on the graph of the function $f(x)=2\sin (\omega x+\phi)$ $(\omega > 0,- \frac{\pi}{2} < \phi < 0 )$, and the terminal side of angle $\phi$ passes through point $P(1,- \sqrt{3} )$. If $|f(x_{1})-f(x_{2})|=4$ when, the minimum value of $|x_{1}-x_{2}|$ is $\frac{\pi}{3}$.

$(1)$ Find the expression of the function $f(x)$;

$(2)$ If the equation $3[f(x)]^{2}-f(x)+m=0$ has two different solutions within $x\in( \frac{\pi}{9} , \frac{4\pi}{9} )$, find the range of the real number $m$.

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