
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 that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$, respectively, and $\overrightarrow{m}=(-1,\sin A)$, $\overrightarrow{n}=(\cos A+1, \sqrt {3})$.
1. Find the maximum value of $f(A)= \overrightarrow{m}\cdot \overrightarrow{n}$.
2. If $\overrightarrow{m} \perp \overrightarrow{n}$, $b= \frac {4 \sqrt {2}}{3}$, $\cos B= \frac { \sqrt {3}}{3}$, find the value of $a$.

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