
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  $T_A=\triangle AA_1A_2.$  Construct triangles  $T_B=\triangle BB_1B_2$  and  $T_C=\triangle CC_1C_2$  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   $$ \dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}. $$   Find  $(b,c).$ 

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