
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 $f(x)=\sqrt{3}\sin{2\omega x}+2\cos^2{\omega x}$, if the distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{\pi}{2}$.  
$(1)$ Find the value of $\omega$ and the interval where $f(x)$ is monotonically decreasing.  
$(2)$ If the equation $f(x)=m$ has a solution in $(-\frac{\pi}{4}, \frac{\pi}{4})$, find the range of real number $m$.

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