
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 vectors $\overrightarrow{a}=(\sin \omega x, \sqrt{3}\cos 2\omega x)$ and $\overrightarrow{b}=(\frac{1}{2}\cos \omega x, \frac{1}{4})$, where $\omega > 0$, $f(x)=\overrightarrow{a} \cdot \overrightarrow{b}$, and the minimum distance between the axes of symmetry of the function $y=f(x)$ is $\frac{\pi}{2}$.  
$(1)$ Find the analytical expression of $f(x)$;  
$(2)$ The equation $f(x)=m$ has exactly two distinct real solutions in the interval $[0, \frac{7\pi}{12}]$. Find the range of real numbers for $m$.

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