
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 $\triangle ABC$, with the opposite sides of the internal angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, it satisfies $\frac{\sqrt{2}b-c}{a} = \frac{\cos C}{\cos A}$  
$(1)$ Find the size of angle $A$;  
$(2)$ If $a=10$, $b=8\sqrt{2}$, and angle $C$ is the smallest, find the area $S$ of $\triangle ABC$.

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