
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 $f(x) = \sin^2 \omega x + \sqrt{3} \cos \omega x \cos \left( \frac{\pi}{2} - \omega x \right)$ $(\omega > 0)$, and the graph of $y=f(x)$ has a distance of $\frac{\pi}{2}$ between two adjacent axes of symmetry.  
$(1)$ Calculate the value of $f\left(\frac{\pi}{6}\right)$.  
$(2)$ If the function $f(kx+\frac{\pi}{12})$ $(k > 0)$ is monotonically increasing on the interval $\left[-\frac{\pi}{6}, \frac{\pi}{3}\right]$, find the range of $k$.

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