
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 $\triangle ABC$, $a=5$, $b=6$.  
$(1)$ If $\cos B=-\frac{4}{5}$, find the values of $A$ and the radius $R$ of the circumcircle of $\triangle ABC$.  
$(2)$ If the area of $\triangle ABC$ is $S=\frac{15\sqrt{7}}{4}$, find the value of $c$.

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