
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 $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$, respectively. Given that $(a + 2c) \cos B + b \cos A = 0$,
$(1)$ Find angle $B$;
$(2)$ If $b = 3$ and the perimeter of triangle $ABC$ is $3 + 2\sqrt{3}$, find the area of $ABC$.

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