Four, (50 points) Let $n$ be a positive integer, and the point set $S$ in three-dimensional space satisfies the following properties:
(1) There do not exist $n$ planes in space such that each point in the point set $S$ is on at least one of these $n$ planes;
(2) For each point $X \in S$, there exist $n$ planes such that each point in $S \backslash\{X\}$ is on at least one of these $n$ planes.
Find the minimum and maximum number of points in the point set $S$.
Four, first find the minimum possible value of $|S|$.
Since a plane can be determined by any three points, we have $|S| \geqslant 3 n+1$.
And when any four points among the $3 n+1$ points are not coplanar, the condition of the problem is satisfied.
Thus, the minimum possible value of $|S|$ is $3 n+1$.
Next, find the maximum possible value of $|S|$.
For each $P=\left(x_{p}, y_{p}, z_{p}\right) \in S$, let the line
$$
l_{i, p}: a_{i, p} x+b_{i, p} y+c_{i, p} z+d_{i, p}=0(1 \leqslant i \leqslant n)
$$

cover $S \backslash\{P\}$.
By the problem's condition,
$P \notin l_{1, p} \cup l_{2, p} \cup \cdots \cup l_{n, p}$.
Let $p_{p}(x, y, z)$
$$
=\prod_{i=1}^{n} \frac{a_{i, p} x+b_{i, p} y+c_{i, p} z+d_{i, p}}{a_{i, p} x_{p}+b_{i, p} y_{p}+c_{i, p} z_{p}+d_{i, p}} .
$$

Then $p_{p}(x, y, z)$ is a ternary $n$-degree polynomial, and
$$
\left.p_{p}\right|_{S \backslash P \mid}=0, p_{p}(P)=1 \text {. }
$$

Thus, $\left\{p_{p} \mid P \in S\right\}$ is linearly independent in $F_{n}\left(F_{n}\right.$ is the vector space of ternary polynomials of degree not exceeding $n$).
Therefore, $|S| \leqslant \operatorname{dim} F_{n}=\mathrm{C}_{n+3}^{3}$.
Now, provide an example where the set $S$ contains $\mathrm{C}_{n+3}^{3}$ points.
As shown in Figure 3, let $U_{n}$ be a tetrahedral point array consisting of $\mathrm{C}_{n+3}^{3}$ points.
$$
\begin{array}{l}
\text { Then }\left|U_{n}\right| \\
=\frac{(n+2)(n+1)}{2}+\frac{(n+1) n}{2}+\cdots+\frac{2 \times 1}{2} \\
=\mathrm{C}_{n+3}^{3} .
\end{array}
$$

For each point $P \in U_{n}$, $U_{n} \backslash\{P\}$ can be covered by $n$ planes. However, $U_{n}$ cannot be covered by $n$ planes.

In summary, the maximum number of points in the set $S$ is $\mathrm{C}_{n+3}^{3}$.