
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
On the sides \( AB, AC \), and \( BC \) of the equilateral triangle \( ABC \), points \( C_{1}, B_{1} \), and \( A_{1} \) are located respectively, such that the triangle \( A_{1} B_{1} C_{1} \) is equilateral. The segment \( BB_{1} \) intersects the side \( C_{1} A_{1} \) at point \( O \), with \( \frac{BO}{OB_{1}}=k \). Find the ratio of the area of triangle \( ABC \) to the area of triangle \( A_{1} B_{1} C_{1} \).

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