
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
Sarah and Hagar play a game of darts. Let \(O_0\) be a circle of radius 1. On the \(n\)th turn, the player whose turn it is throws a dart and hits a point \(p_n\) randomly selected from the points of \(O_{n-1}\). The player then draws the largest circle that is centered at \(p_n\) and contained in \(O_{n-1}\), and calls this circle \(O_n\). The player then colors every point that is inside \(O_{n-1}\) but not inside \(O_n\) her color. Sarah goes first, and the two players alternate turns. Play continues indefinitely. If Sarah's color is red, and Hagar's color is blue, what is the expected value of the area of the set of points colored red?

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