
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
In triangle $\triangle ABC$, the sides opposite to $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Let $\overrightarrow{m}=(\cos A, -\sin A)$, $\overrightarrow{n}=(\cos C, \sin C)$, and $\overrightarrow{m} \cdot \overrightarrow{n}=-\frac{1}{3}$. <br/>$(1)$ Find the value of $\cos B$; <br/>$(2)$ If $c=2$ and the area of $\triangle ABC$ is $2\sqrt{2}$, find the length of side $b$.

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