
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 triangle $ABC$ with opposing sides of interior angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is known that $a + 2c = 2b \cos A$.  
$(1)$ Find the magnitude of angle $B$;  
$(2)$ If $b = 2\sqrt{3}$ and $a+c=4$, find the area of the triangle $ABC$.

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