
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 $\overrightarrow{a} = (\sin\alpha, \cos\alpha)$ and $\overrightarrow{b} = (6\sin\alpha + \cos\alpha, 7\sin\alpha - 2\cos\alpha)$, let the function $f(\alpha) = \overrightarrow{a} \cdot \overrightarrow{b}$.
(1) Find the maximum value of the function $f(\alpha)$;
(2) In an acute triangle ABC, where the sides opposite angles A, B, and C are denoted as a, b, and c respectively, $f(A) = 6$, and the area of $\triangle ABC$ is 3, with $b+c=2+3\sqrt{2}$. Find the value of a.

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