
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 \( A_{1}, B_{1}, \) and \( C_{1} \) be the points on the sides of the triangle \(ABC\) such that

\[ 
\frac{A C_{1}}{C_{1} B} = \frac{B A_{1}}{A_{1} C} = \frac{C B_{1}}{B_{1} A} = \lambda 
\]

Let \( A_{2}, B_{2}, \) and \( C_{2} \) be the points on the sides of the triangle \(A_{1} B_{1} C_{1}\) such that

\[ 
\frac{A_{1} C_{2}}{C_{2} B_{1}} = \frac{B_{1} A_{2}}{A_{2} C_{1}} = \frac{C_{1} B_{2}}{B_{2} A_{1}} = \lambda 
\]

For what value of \( \lambda \) will the area of triangle \( A_{2} B_{2} C_{2} \) be half the area of triangle \(ABC\)?

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