
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 $f(x)=2\sin(\omega x+\phi -\frac{\pi}{6})$ $(0 < \phi < \pi,\omega > 0)$ is an even function, and the distance between two adjacent axes of symmetry of the graph of $y=f(x)$ is $\frac{\pi}{2}$.

$(1)$ Find the value of $f\left( \frac{\pi}{8} \right)$;

$(2)$ Find the equation of the axis of symmetry for the function $y=f\left( x+\frac{\pi}{6} \right)$;

$(3)$ When $x\in \left[ 0,\frac{7\pi}{12} \right]$, the equation $f(x)=m$ has two different real roots, find the range of $m$.

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