Convex Hull Surface Area

A three-dimensional shape is convex if, for any two points inside it, the line segment joining them lies entirely within the shape. For a set of points $X$ in three-dimensional space, the convex hull of $X$ is the smallest convex shape that contains every point of $X$.

For example, let $X = \{(0,0,0),\ (10,0,0),\ (0,10,0),\ (0,0,10)\}$. The convex hull of $X$ is the tetrahedron whose vertices are the points of $X$. This tetrahedron contains the point $(1,1,1)$, so adding $(1,1,1)$ to $X$ would not change the hull.

Given $X$, compute the surface area of its convex hull, rounded to the nearest integer.

Note: every face of a convex hull is a polygon. In this problem you may assume that at most $3$ points of $X$ lie on any single face of the hull (so every face is a triangle).

The input contains multiple test cases. Each test case begins with an integer $n$ ($4 \le n \le 25$), the number of points in $X$. The next $n$ lines each contain three integers: the $x$, $y$, and $z$ coordinates of one point. Every coordinate satisfies $-100 \le x, y, z \le 100$.

The input ends with a line containing $n = 0$, which must not be processed.

For each test case, print a single line containing the surface area of the convex hull, rounded to the nearest integer (for example, $2.499$ rounds to $2$, while $2.5$ rounds to $3$).

To avoid ambiguity from rounding, the test data is built so that every answer is at least $0.001$ away from a rounding boundary (for instance, an area is never exactly $2.4997$).