
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 function $f(x) = a\sin(\omega x) + b\cos(\omega x)$ ($\omega > 0$) defined on $\mathbb{R}$ has a period of $\pi$, and for all $x \in \mathbb{R}$, $f(x) \leq f\left( \frac{\pi}{12} \right) = 8$.

(1) Find the expression for the function $f(x)$;  
(2) If $g(x) = f\left( \frac{\pi}{6} - x \right)$, find the interval where $g(x)$ is decreasing.

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