
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 $f(x)=\sin(\omega x+\phi)$ $(\omega>0, |\phi|\leq\frac{\pi}{2})$ with a center of symmetry at $(-\frac{\pi}{4},0)$, an axis of symmetry at $x=\pi$, and a monotonically decreasing interval $[\pi, \frac{3\pi}{2}]$, find the sum of all values of $\omega$ that satisfy these conditions.

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