
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 $f(x)=2\sin(\omega x+\frac{5\pi}{12})+\cos(\omega x-\frac{\pi}{12})$ $(\omega>0)$, and $f(\pi)=3$. The graph of $f(x)$ on the interval $(0,\frac{\pi}{12})$ intersects the line $y=\frac{3\sqrt{3}}{2}$ at exactly $2$ points, find the possible values of $\omega$.

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