
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 a triangle $ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively, and the relationship $b \tan B = \sqrt{3} (a \cos C + c \cos A)$.
$(1)$ Find the value of angle $B$.
$(2)$ If the area of triangle $ABC$ is $\frac{7\sqrt{3}}{3}$ and $a + c = 8$, find the length of side $b$.

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