
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and it is given that $bc=5$ and $\cos \frac{A}{2}=\frac{3\sqrt{10}}{10}$.

$(1)$ Find the area of $\triangle ABC$;

$(2)$ If $\sin B=5\sin C$, find the values of $a$, $b$, $c$.

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