
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
Let \( \triangle ABC \) be an equilateral triangle with side length 1. Points \( A_1 \) and \( A_2 \) are chosen on side \( BC \), points \( B_1 \) and \( B_2 \) are chosen on side \( CA \), and points \( C_1 \) and \( C_2 \) are chosen on side \( AB \) such that \( BA_1 < BA_2 \), \( CB_1 < CB_2 \), and \( AC_1 < AC_2 \).

Suppose that the three line segments \( B_1C_2 \), \( C_1A_2 \), and \( A_1B_2 \) are concurrent, and the perimeters of triangles \( AB_2C_1 \), \( BC_2A_1 \), and \( CA_2B_1 \) are all equal. Find all possible values of this common perimeter.

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