
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 the conditions of the previous problem, let the points \( A_{12} \), \( A_{23} \), \( A_{34} \), and \( A_{41} \) divide the sides of the quadrilateral in the ratios \(\lambda_{12}\), \(\lambda_{23}\), \(\lambda_{34}\), \(\lambda_{41}\), respectively. In what ratio is each of the segments \( A_{12}A_{34} \) and \( A_{23}A_{41} \) divided by the point of their intersection?

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