
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 \( A_1 A_2 A_3 A_4 A_5 A_6 \) be a regular hexagon. Each of its diagonals \( A_{i-1} A_{i+1} \) is divided into the same ratio \( \frac{\lambda}{1-\lambda} \), where \( 0<\lambda<1 \), by a point \( B_i \) in such a way that \( A_i, B_i \), and \( B_{i+2} \) are collinear (\( i \equiv 1, \ldots, 6 \pmod{6} \)). Compute \( \lambda \).

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