
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 $a > 0$, the function $f(x)=-2a\sin\left(2x+\frac{\pi}{6}\right)+2a+b$, where $x \in \left[0, \frac{\pi}{2}\right]$, satisfies $-5 \leq f(x) \leq 1$.
$(1)$ Find the constants $a$ and $b$;
$(2)$ Let $g(x)=f\left(x+\frac{\pi}{2}\right)$ and $\lg[g(x)] > 0$, determine the intervals of monotonicity for $g(x)$.

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