
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  $C_B$  at  $C_1$  and  $C_2$ , respectively.
Find the perimeter of the triangle formed by lines  $A_1A_2$ ,  $B_1B_2$ ,  $C_1C_2$ .

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