
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 $f(x) = 2\cos x \sin\left(x - \frac{\pi}{6}\right) + \frac{1}{2}$.  
(1) Find the equation of the axis of symmetry for the function $f(x)$.  
(2) If the equation $\sin 2x + 2|f\left(x + \frac{\pi}{12}\right)| - m + 1 = 0$ has three real solutions for $x \in \left[-\frac{\pi}{3}, \frac{\pi}{2}\right]$, find the range of the real number $m$.

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