
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, a, b, c are the lengths of sides opposite to angles A, B, C, respectively. Triangle ABC satisfies three of the following four conditions at the same time:①a<sup>2</sup>+c<sup>2</sup>=b<sup>2</sup>-$\frac{{2\sqrt{3}}}{3}$ac; ②1+cos2A=2sin<sup>2</sup>$\frac{A}{2}$; ③a=$\sqrt{3}$; ④b=2．<br />(1) What are the possible combinations of conditions that allow triangle ABC to have a solution?<br />(2) Choose one combination from (1) and find the area of triangle ABC.

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