
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 $A(x_{1},f(x_{1})), B(x_{2},f(x_{2}))$ on the graph of the function $f(x)=2\sin(\omega x+\varphi)$ $(\omega > 0, -\frac{\pi}{2} < \varphi < 0)$, and the terminal side of angle $\varphi$ 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 analytical expression of the function $f(x)$;

$(2)$ When $x \in [0,\frac{2\pi}{3}]$, find the intervals of monotonic increase for the function $f(x)$;

$(3)$ When $x \in \left[ 0,\frac{\pi}{6} \right]$, the inequality $mf(x)+2m \geqslant f(x)$ holds true for all times, find the range of the real number $m$.

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