
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
Given a triangle $ABC$ with sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and it is given that $(3b-c)\cos A = a\cos C$.
(1) Find the value of $\cos A$;
(2) If the area of $\triangle ABC$ is $S=2\sqrt{2}$, find the minimum value of the perimeter of $\triangle ABC$.

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