Shrinking Inscribed Polygon

A polygon is said to be inscribed in a circle when all of its vertices lie on that circle. Given a polygon inscribed in a circle, determine the minimum number of vertices that must be removed so that the remaining polygon becomes a regular polygon. A regular polygon is a polygon in which every side has the same length and every interior angle has the same measure.

To remove a vertex $v$ from the polygon, first find its two neighboring vertices $w_1$ and $w_2$, then connect $w_1$ and $w_2$ with a new edge. Doing so merges the two arcs that had $v$ between them into a single arc.

For example, from an inscribed polygon with $10$ vertices you can remove a suitable set of $5$ vertices to form a regular pentagon.

The polygon always has at least $3$ sides.

The input consists of several test cases.

The first line of each test case contains the number of vertices $N$ of the inscribed polygon. ($3 \le N \le 10^4$) The second line contains $N$ integers $X_i$. ($1 \le X_i \le 10^3$)

$X_i$ is the arc length between vertex $i$ and vertex $(i+1) \bmod N$, given in clockwise order. Note that an arc is measured along the circle's circumference, not as a chord.

The last line of the input contains a single $0$, which is not processed.

For each test case, print on one line the minimum number of vertices that must be removed to form a regular polygon. If no regular polygon can be formed, print $-1$.