
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 $a$, $b$, and $c$ respectively. Given that $(a-b\cos C)\sin B+\sqrt{3}b\cos B\cos C=0$, $b=3$, and $B\neq\frac{\pi}{2}$.  
$(1)$ If $a=5$, find $c$;  
$(2)$ If the length of the median on side $AB$ is $\frac{7}{2}$, find the area of $\triangle ABC$.

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