
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 $f(x)=\sin(\omega x+\varphi)$ ($\omega>0$, $|\varphi|<\frac{\pi}{2}$), the graph of which is symmetric about $x=\frac{\pi}{4}$, $f(-\frac{\pi}{4})=0$, and $f(x)$ has exactly $3$ local maximum points in the interval $(0,\pi)$, determine the value of $\omega$.

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