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 $A_{500}A_{1500}A_{2500}A_{3500}$?

[i]Proposed by Evan Chen[/i]