Angry Cows

Bessie built a game called "Angry Cows". A slingshot launches one cow onto a number line, and the goal is to detonate every hay bale on that line.

There are $N$ hay bales at distinct integer positions $x_1, x_2, \ldots, x_N$. A cow launched with power $R$ that lands at position $x$ detonates every hay bale in the range $x - R$ to $x + R$. Those bales then all explode at the same time with blast radius $R - 1$, and every bale caught in one of those blasts that has not exploded yet explodes at the same time with blast radius $R - 2$. The chain continues until no new bale explodes or the radius drops below $0$.

The landing position does not have to be an integer. Find the smallest power $R$ for which some landing position detonates all $N$ bales.

The first line contains $N$ ($2 \le N \le 50\,000$).

Each of the next $N$ lines contains one position $x_i$ ($0 \le x_i \le 1\,000\,000\,000$). All positions are distinct, and they are not necessarily sorted.

Print the smallest power $R$ that detonates every hay bale, with exactly one digit after the decimal point. The answer is always a multiple of $0.5$.

Suppose the bales sit at $1, 3, 8, 10, 11$. A cow launched with power $3$ that lands at position $5$ detonates the bales at $3$ and $8$. Those two explode together with radius $2$ and catch the bales at $1$ and $10$, which explode together with radius $1$ and catch the bale at $11$. The last bale explodes with radius $0$.