
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 angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it is given that $\frac{2a}{\cos A} = \frac{3c-2b}{\cos B}$.
(1) If $b = \sqrt{5}\sin B$, find $a$;
(2) If $a = \sqrt{6}$ and the area of $\triangle ABC$ is $\frac{\sqrt{5}}{2}$, find $b + c$.

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