
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
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, and it is known that $\cos B = \frac{1}{3}$.  
$(1)$ Find the value of $\cos^2\frac{B}{2} + \tan^2\frac{A+C}{2}$.  
$(2)$ If $b=4$ and the area of $\triangle ABC$ is $2\sqrt{2}$, find the value of $c$.

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