
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  $A_1A_2 ... A_8$  be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each  $i = 1, ... , 8$ , define  $B_i$  as the intersection between segments  $A_iA_{i+4}$  and  $A_{i-1}A_{i+1}$ , where  $A_{j+8} = A_j$  and  $B_{j+8} = B_j$  for all  $j$ . Show some number  $i$ , amongst 1, 2, 3, and 4 satisfies
\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]

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