Rope Folding

Farmer John has a rope of length $L$ ($1 \le L \le 10{,}000$) that he uses for tasks around his farm. The rope has $N$ knots ($1 \le N \le 100$) tied into it at distinct integer positions, including one knot at each of its two endpoints (positions $0$ and $L$).

FJ can fold the rope back onto itself at certain points. When he folds at a point, one side of the rope reflects over onto the other side and the two strands overlap. A fold is good if, wherever the two strands lie on top of each other, every knot on one strand lines up exactly with a knot on the other strand.

Formally, a fold at position $f$ ($0 < f < L$) reflects a knot at position $x$ to position $2f - x$. Let the shorter side have length $d = \min(f,\, L - f)$; the two strands then overlap on the interval $[f - d,\, f + d]$. The fold is good if, for every knot $x$ inside that overlap interval, its mirror position $2f - x$ is also a knot.

Folding exactly at a knot is allowed, but folding at either endpoint is not. Extra knots on the longer side of the fold — outside the overlap interval — do not matter. FJ only ever makes a single fold at a time.

Count the number of positions at which FJ can make a good fold.

• Line 1: Two space-separated integers, $N$ and $L$.
• Lines $2 \ldots N+1$: Each line contains one integer in the range $0 \ldots L$, the position of a single knot. Two of these positions are always $0$ and $L$.

• Line 1: A single integer — the number of positions at which a good fold can be made.

For example, if the rope has length $L = 10$ with knots at $0, 2, 4, 6, 10$, the four good fold positions are $1, 2, 3,$ and $8$:

• Fold at $1$: the overlap is $[0, 2]$; knots $0$ and $2$ mirror onto each other.
• Fold at $2$: the overlap is $[0, 4]$; knots $0, 2, 4$ are symmetric about $2$.
• Fold at $3$: the overlap is $[0, 6]$; knots $0, 2, 4, 6$ are symmetric about $3$.
• Fold at $8$: the overlap is $[6, 10]$; knots $6$ and $10$ mirror onto each other (the extra knots $0, 2, 4$ sit on the longer side and are ignored).

A fold position may be a half-integer: since the endpoint knot on the shorter side must reflect onto an integer knot, every good fold occurs where $2f$ is an integer.