
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
Given an acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is known that $b+c=10$, $a= \sqrt{10}$, $5b\sin A\cos C+5c\sin A\cos B=3 \sqrt{10}$.  
(1) Find the cosine value of $A$;  
(2) Find the values of $b$ and $c$.

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