
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, $c$ respectively, and $b=a(\sin C+\cos C)$.  
$(1)$ Find angle $A$.  
$(2)$ Among the three conditions: $\left(1\right)a=2$, $\left(2\right)B=\frac{π}{3}$, $\left(3\right)c=\sqrt{2}b$, choose two conditions that uniquely determine $\triangle ABC$ and solve them. If ____, ____, find the area of $\triangle ABC$.  
Note: If multiple conditions are chosen and solved separately, the first solution will be scored.

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