
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 triangle $A_{1} A_{2} A_{3}$, the point $B_{i}$ is the quarter-point on the side $A_{i} A_{i+1}$ closest to $A_{i}$, and the point $C_{i}$ is the quarter-point on the side $A_{i} A_{i+1}$ closest to $A_{i+1}$ $(i=1,2,3, A_{4} \equiv A_{1})$. What fraction of the area of triangle $A_{1} A_{2} A_{3}$ is the area of the common region of triangles $B_{1} B_{2} B_{3}$ and $C_{1} C_{2} C_{3}$?

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