
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 the conditions ① $\sqrt{2}c=asinC+ccosA$, ② $sin({B+C})=\sqrt{2}-1+2sin^2\frac{A}{2}$, ③ $\sqrt{2}cos({\frac{π}{2}-A})=sin2A$, choose any one of these three conditions as the known condition, and then answer the following questions. Let the sides opposite to the internal angles $A$, $B$, $C$ of triangle $\triangle ABC$ be $a$, $b$, $c$, and the area of $\triangle ABC$ be $S$. Given _____.  
$(1)$ Find $A$;  
$(2)$ If $S=6$ and $b=2\sqrt{2}$, find $a$.

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