
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 $f(x) = \sqrt{3} \sin{\omega x} \cdot \cos{\omega x} + \cos^{2}{\omega x} - \frac{1}{2} (\omega > 0)$, the distance between two adjacent symmetry axes is $\frac{x}{2}$.

(1) Find the value of $\omega$;

(2) Translate the graph of the function $f(x)$ to the left by $\frac{x}{6}$ units, then stretch the horizontal coordinates of all points on the obtained graph to twice their original length (while keeping the vertical coordinates unchanged) to obtain the graph of function $y = g(x)$. If the function $y = g(x) - k$ has a zero in the interval $[-\frac{x}{6}, \frac{2\pi}{3}]$, find the range of values for the real number $k$.

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