
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 vectors $\overrightarrow{m}=(a,b)$ and $\overrightarrow{n}=(\sin 2x, 2\cos ^{2}x)$, let $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$. If $f(0)=8$ and $f(\frac{\pi }{6})=12$,

(1) Find the values of $a$ and $b$;

(2) Determine the maximum value of the function $f(x)$ and the set of $x$ values for which this maximum is attained;

(3) Find the interval(s) on which the function $f(x)$ is strictly increasing.

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