
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 $\triangle ABC$, $A+B=5C$, $\sin \left(A-C\right)=2\sin B$.  
$(1)$ Find the value of $A$;  
$(2)$ If $CM=2\sqrt{7}$ and $M$ is the midpoint of $AB$, find the area of $\triangle ABC$.

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