
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 $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively, with $a=7$ and $2c\cos B=(3a-2b)\cos C$. Find:<br/>
$(1)$ $\cos C$;<br/>
$(2)$ If $B=2C$ and $M$ is the incenter of $\triangle ABC$, find the area of $\triangle AMC$.

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