
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 denoted by $a$, $b$, and $c$ respectively, and it is given that $2c^2-2a^2=b^2$.  
(1) Find the value of $\frac{c \cos A - a \cos C}{b}$;  
(2) If $a=1$ and $\tan A = \frac{1}{3}$, find the area $S$ of triangle $ABC$.

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