AMO-Bench 2025 Problem 38

In a $d$-dimensional Euclidean space, there are two definitions:

1. Encapsulation: For a point $p$, if there exists a ball centered at $p$ with sufficiently small radius such that every point within this ball (except $p$ itself) belongs to at least one convex set $K_i$ (where $i=1, \dots, n$), then we say point $p$ is encapsulated by these convex sets;
2. Contact: For a set $K$ and a point $p$, if $p$ does not lie inside $K$ but any neighborhood of $p$ (no matter how small) contains points from $K$, we say point $p$ contacts set $K$ (This is equivalent to saying $p$ is a boundary point of $K$). Let $K_1, K_2, K_3$ be three convex sets in $d$-dimensional space. Let $S$ be the set of points encapsulated by these three convex sets. The maximum possible value of $|S|$ is a function theat depends only on the dimension $d$. Based on the above definitions and known conditions, the maximum value can be derived. Let $f(d, 3) = \max |S|$. Please provide the expression for $f(d, 3)$: $f(d, 3) =$