
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 $\triangle ABC$, $a=1$, $b=2$.  
$(1)$ If $c=2\sqrt{2}$, find the area of $\triangle ABC$.  
$(2)$ Choose one of the following three conditions as the given information to ensure the existence of $\triangle ABC$, and find $\angle A$.  
Condition ①: $\angle B=2\angle A$; Condition ②: $\angle B=\frac{π}{3}+\angle A$; Condition ③: $\angle C=2\angle A$.  
Note: If the chosen condition does not meet the requirements, 0 points will be awarded for question (2); if multiple valid conditions are chosen and answered separately, the first answer will be scored.

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