
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 $ABC$, let the sides opposite to angles $A$, $B$, and $C$ be denoted by $a$, $b$, and $c$ respectively, with the condition that $a\cos B = (2c - b)\cos A$.  
$(1)$ Find the measure of angle $A$.  
$(2)$ Suppose $b=3$, and let point $M$ be on segment $BC$ such that $\overrightarrow{AB} + \overrightarrow{AC} = 2\overrightarrow{AM}$ and $|\overrightarrow{AM}| = \frac{3\sqrt{7}}{2}$. Determine the area of $\triangle ABC$.

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