
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 $f(x) = a \sin(2x - \frac{\pi}{3}) + b$.

(1) If $a > 0$, write out the monotonically decreasing interval and the symmetry center of the function $f(x)$;

(2) If $x \in [0, \frac{\pi}{2}]$, the minimum value of the function $f(x)$ is $-2$, and the maximum value is $\sqrt{3}$, find the maximum value of the function $g(x) = a \cos^2 x + \sin x + b$.

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