
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
Given the convex regular pentagon \( A_1 A_2 A_3 A_4 A_5 \), determine the point \( B_1 \) on the line \( A_1 A_3 \) such that, denoting the intersection of the lines \( B_1 A_2 \) and \( A_3 A_5 \) by \( B_3 \),

$$
\frac{A_1 B_1}{B_1 A_3} \quad \text{and} \quad \frac{A_3 B_3}{B_3 A_5}
$$

are equal with sign considered.

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