
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 $f\left( x \right)=\sin^{2}{\omega x}+\sqrt{3}\cos {\omega x}\cos {\left( \frac{\pi }{2}-\omega x \right)}\left( \omega > 0 \right)$, and the distance between adjacent symmetry axes of the graph of the function $y=f(x)$ is $\frac{\pi }{2}$.
(1) Find the value of $f\left( \frac{\pi }{6} \right)$.
(2) If the function $f\left( kx+\frac{\pi }{12} \right)\left( k > 0 \right)$ is monotonically increasing in 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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