
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 \(ABC\), points \(C_{1}\), \(B_{1}\), and \(A_{1}\) are located on sides \(AB\), \(AC\), and \(CB\), respectively, such that

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

Let the radii of the incircles of triangles \(AC_{1}B_{1}\), \(BC_{1}A_{1}\), \(CA_{1}B_{1}\), \(A_{1}B_{1}C_{1}\), and \(ABC\) be \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r\) respectively. For what values of \(\lambda\) does the following equation hold?

\[ \frac{1}{r_{1}} + \frac{1}{r_{2}} + \frac{1}{r_{3}} = \frac{1}{r_{4}} + \frac{4}{r} \]

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