
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 $\sin 2x+2|f(x+ \frac {\pi}{12})|-m+1=0$ 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}
user