
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  $A_1A_2 \dots A_{4000}$  be a regular  $4000$ -gon. Let  $X$  be the foot of the altitude from  $A_{1986}$  onto diagonal  $A_{1000}A_{3000}$ , and let  $Y$  be the foot of the altitude from  $A_{2014}$  onto  $A_{2000}A_{4000}$ . If  $XY = 1$ , what is the area of square  $A_{500}A_{1500}A_{2500}A_{3500}$ ?

*Proposed by Evan Chen*

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