
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 $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$, respectively. It is given that $b(3b-c)\cos A = \overrightarrow{CA} \cdot \overrightarrow{CB}$.

(1) Find the value of $\cos A$;  
(2) If the area of $\triangle ABC$ is $2\sqrt{2}$, and the length of the median $CM$ from $C$ to side $AB$ is $\frac{\sqrt{17}}{2}$, find the lengths of $b$ and $c$.

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