
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
Plot points \(A, B, C\) at coordinates \((0,0), (0,1)\), and \((1,1)\) in the plane, respectively. Let \(S\) denote the union of the two line segments \(AB\) and \(BC\). Let \(X_1\) be the area swept out when Bobby rotates \(S\) counterclockwise 45 degrees about point \(A\). Let \(X_2\) be the area swept out when Calvin rotates \(S\) clockwise 45 degrees about point \(A\). Find \(\frac{X_1 + X_2}{2}\).

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