
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 $a=(\sin \alpha, \cos \alpha)$ and $b=(6\sin \alpha + \cos \alpha, 7\sin \alpha - 2\cos \alpha)$, define function $f(\alpha)=ab$.

1. Find the maximum value of function $f(\alpha)$.
2. In an acute triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(A)=6$, the area of $\triangle ABC$ is $3$, and $b+c=2+3\sqrt{2}$, find the value of $a$.

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