
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
The right triangle \(ABC\) has legs \(|AB| = 5 \text{ cm}\) and \(|BC| = 12 \text{ cm}\). The vertices of the triangle \(ABC\) are transformed as follows:

- Point \( A \) is mapped to point \( A' \) by reflection over the angle bisector of angle \(ABC\),
- Point \( B \) is mapped to point \( B' \) by reflection over the midpoint of segment \(BC\),
- Point \( C \) is mapped to point \( C' \) by reflection over the perpendicular bisector of segment \(AB\).

Determine the area of triangle \( A'B'C' \).

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