
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
Let \( S \) be the set of points \((x, y)\) in the plane such that the sequence \( a_n \) defined by \( a_0 = x \), \( a_{n+1} = \frac{a_n^2 + y^2}{2} \) converges. What is the area of \( S \)?

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