
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 that $A(x\_1,y\_1)$ and $B(x\_2,y\_2)$ are any two points on the function $f(x)=2\sin (wx+φ) (w > 0, - \frac {π}{2} < φ < 0)$, and the terminal side of angle $φ$ passes through point $P(1,- \sqrt {3})$. If $|f(x\_1)-f(x\_2)|=4$ when $|x\_1-x\_2|$'s minimum value is $\frac {π}{3}$,

1. Find the analytical expression of function $f(x)$;
2. Find the increasing interval of function $f(x)$;
3. Find the range of real number $m$ such that the inequality $mf(x)+2m\geqslant f(x)$ always holds when $x\in[0, \frac {π}{6}]$.

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