
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 $a$, $b$, $c$ are the lengths of the sides opposite to the angles of $\Delta ABC$ respectively, and $\sqrt{3}\cos C + \sin C = \frac{\sqrt{3}a}{b}$.

$(1)$ Find the magnitude of angle $B$;

$(2)$ If $a + c = 5\sqrt{7}$ and $b = 7$, find the value of $\overrightarrow{AB} \cdot \overrightarrow{BC}$.

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