
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 $\overrightarrow{n} = (\sin 2x, 2\cos^2 x)$, if $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$, and $f(0) = 8, f\left( \frac{\pi}{6} \right) = 12$.

(1) Find the values of $a$ and $b$;  
(2) Find the maximum value of the function $f(x)$ and the set of $x$ values for which this maximum is achieved;  
(3) Find the intervals on which $f(x)$ is increasing.

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