
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 $f(x)=\sin \omega x \cdot \cos \omega x + \sqrt{3} \cos^2 \omega x - \frac{\sqrt{3}}{2}$, where $\omega > 0$, the lines $x=x_{1}$ and $x=x_{2}$ are any two symmetric axes of the graph of $y=f(x)$, and the minimum value of $|x_{1}-x_{2}|$ is $\frac{\pi}{4}$.  
$(1)$ Find the expression of $f(x)$;  
$(2)$ Shift the graph of the function $f(x)$ to the right by $\frac{\pi}{8}$ units, then stretch the horizontal coordinates of all points on the resulting graph by a factor of $2$, while keeping the vertical coordinates unchanged, to obtain the graph of the function $y=g(x)$. If the equation $g(x)+k=0$ has exactly one real number solution in the interval $[0,\frac{\pi}{2}]$, find the range of real numbers for $k$.

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