
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 \( ABC \) be an equilateral triangle of side length 1. For a real number \( 0 < x < 0.5 \), let \( A_1 \) and \( A_2 \) be the points on side \( BC \) such that \( A_1B = A_2C = x \), and let \( T_A = \triangle A A_1 A_2 \). Construct triangles \( T_B = \triangle B B_1 B_2 \) and \( T_C = \triangle C C_1 C_2 \) similarly. There exist positive rational numbers \( b, c \) such that the region of points inside all three triangles \( T_A, T_B, T_C \) is a hexagon with area
\[
\frac{8 x^2 - b x + c}{(2-x)(x+1)} \cdot \frac{\sqrt{3}}{4}.
\]

Find \( (b, c) \).

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