
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 $\overrightarrow{m}=(\sin x,\cos x),$ and $\overrightarrow{n}=(6\sin x+\cos x,7\sin x-2\cos x)$, define the function $f\left(x\right)= \overset{→}{m}⋅ \overset{→}{n}$.

1. Find the intervals on $\mathbb{R}$ where $f(x)$ is strictly increasing.
2. In an acute triangle $\Delta ABC$, let $a$, $b$, $c$ be the lengths of the sides opposite to angles $A$, $B$, $C$ respectively. If $f(A)=6$ and the area of $\Delta ABC$ is $3$ with $b+c=2+3\sqrt{2}$, find the value of $a$.

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