
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 $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Let vectors $\overrightarrow{m} = (a, c)$ and $\overrightarrow{n} = (\cos C, \cos A)$.
(1) If $\overrightarrow{m} \parallel \overrightarrow{n}$ and $c = \sqrt{3}a$, find angle $A$;
(2) If $\overrightarrow{m} \cdot \overrightarrow{n} = 3b \sin B$ and $\cos A = \frac{4}{5}$, find the value of $\cos C$.

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