
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 $f(x)=\cos(2x+\frac{π}{3})$.  
$(1)$ Find the center of symmetry of the graph of the function $f(x)$ and the interval where the function is monotonically decreasing.  
$(2)$ If $\beta \in (0,\pi)$ and $f(\frac{\beta}{2})=-\frac{1}{2}$, find the value of angle $\beta$.

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