
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 $f(x)=A\sin (3x+\varphi)$ reaches its maximum value of $4$ at $x= \frac{\pi}{12}$, where $A > 0$, $0 < \varphi < \pi$.

$(1)$ Find the intervals of increase for the function $f(x)$;

$(2)$ If $f\left(a+ \frac{\pi}{12}\right)= \frac{12}{5}$, find the value of $\cos (3\alpha+\pi)$.

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