
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 $ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $\cos 2C = -\frac{1}{4}$ and $0 < C < \frac{\pi}{2}$.

(1) Find the value of $\cos C$.
(2) Given $a = 2$ and $2\sin A = \sin C$, find the lengths of $b$ and $c$.

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