
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 points $A(x_1, y_1)$ and $B(x_2, y_2)$ lie on the function $f(x) = 2\sin(wx + \varphi)$ $(w > 0, -\frac{\pi}{2} < \varphi < 0)$, and the terminal side of angle $\varphi$ passes through point $P(1, -\sqrt{3})$. If $|f(x_1) - f(x_2)| = 4$ when $|x_1 - x_2|$ has the minimum possible value of $\frac{\pi}{3}$,  
(1) Find the analytical representation of the function $f(x)$.  
(2) Determine the intervals where the function $f(x)$ is increasing.  
(3) If $x \in \left[0, \frac{\pi}{6}\right]$, the inequality $mf(x) + 2m \geq f(x)$ holds true for all values of $x$ in the interval. Find the range of the real number $m$.

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