
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $\overrightarrow{m}=(2,1)$ and $\overrightarrow{n}=(c\cos C,a\cos B+b\cos A)$, and $\overrightarrow{m} \perp \overrightarrow{n}$.

$(1)$ Find $C$;

$(2)$ If $c^{2}=7b^{2}$ and the area of $\triangle ABC$ is $2\sqrt{3}$, find the value of $b$.

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