
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 \(ABC,\) \(AB = 10,\) \(BC = 11,\) \(AC = 12,\) and point \(P\) is the midpoint of side \(BC.\) Points \(A',\) \(B',\) and \(C',\) are the images of \(A,\) \(B,\) and \(C,\) respectively, after a \(90^\circ\) rotation about \(P.\) What is the area of the union of the two regions enclosed by the triangles \(ABC\) and \(A'B'C'?\)

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