
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 $a$, $b$, and $c$ respectively. It is given that $c-b=2b\cos A$.

$(1)$ If $a=2\sqrt{6}$ and $b=3$, find the value of $c$.

$(2)$ If $\angle C=\frac{\pi}{2}$, find the measure of angle $B$.

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