
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 \(ABC\), let the parabola with focus \(A\) and directrix \(BC\) intersect sides \(AB\) and \(AC\) at \(A_1\) and \(A_2\), respectively. Similarly, let the parabola with focus \(B\) and directrix \(CA\) intersect sides \(BC\) and \(BA\) at \(B_1\) and \(B_2\), respectively. Finally, let the parabola with focus \(C\) and directrix \(AB\) intersect sides \(CA\) and \(CB\) at \(C_1\) and \(C_2\), respectively. If triangle \(ABC\) has sides of length 5, 12, and 13, find the area of the triangle determined by lines \(A_1C_2\), \(B_1A_2\), and \(C_1B_2\).

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