
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 equilateral triangle \( ABC \) with side length 2, 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.

Find the perimeter of the triangle formed by lines \( A_1 A_2, B_1 B_2, C_1 C_2 \).

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