Problem Analysis #26

Let $O_0: x^2 + y^2 = 1$be the unit circle, and let $P(x_0, y_0)$ be a point on $O_0$ with positive $x$-coordinate.Let $Q$ be the intersection of the line tangent to $O_0$ at $P$ and the line $y = -1$.Let $P' = (0,-1)$. Among all circles tangent simultaneously to segment $PQ$, the line$y=-1$, and the circle $O_0$, choose the one lying inside triangle $P'PQ$ and havingradius smaller than that of $O_0$. Call this circle $O_1$.Similarly, for each circle $O_n$ $(n \ge 1)$, define $O_{n+1}$ to be the circle tangent tosegment $PQ$, the line $y = -1$, and the circle $O_n$, lying inside triangle $P'PQ$,and having radius smaller than that of $O_n$.If the sum of the areas of the circles$O_1, O_2, O_3, \dots$is$\frac{\pi}{8},$compute the value of$2024 \cdot x_0^2 \cdot y_0.$oindent