
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 $a$, $b$, $c$ respectively. Let $\vec{m} = (a, c)$ and $\vec{n} = (\cos C, \cos A)$.
(1) If $\vec{m} \parallel \vec{n}$ and $a = \sqrt{3}c$, find angle $A$;
(2) If $\vec{m} \cdot \vec{n} = 3b \sin B$ and $\cos A = \frac{3}{5}$, find the value of $\cos C$.

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