LED

A light-emitting diode (LED) is a semiconductor light source that emits light when an electric current with a voltage higher than a threshold is applied to its leads. ACM R&D recently reported that they have successfully developed a new LED, called ACMOLED. The intensity of light from an ACMOLED changes in two steps as the voltage increases, as shown in the graph below.

An ACMOLED does not emit light for voltages from 0 to $V_1$. It emits light with intensity $L_1 \ge 0$ when the voltage reaches the first threshold $V_1$, and light with intensity $L_2 \ge L_1$ when the voltage reaches the second threshold $V_2$. Formally, let $F(v)$ be the function that maps voltage $v$ to the light intensity. For real numbers $L_1$, $L_2$, $V_1$, and $V_2$ with $0 \le L_1 \le L_2$ and $0 < V_1 < V_2$:

$F(v) = \begin{cases} 0 & \text{if } 0 \le v < V_1 \\ L_1 & \text{if } V_1 \le v < V_2 \\ L_2 & \text{if } v \ge V_2 \end{cases}$

ACM R&D does not know the exact values of $V_1$, $V_2$, $L_1$, and $L_2$. They plan to estimate these four values through repeated experiments. Each experiment applies a current at a chosen voltage and observes the light intensity. After $n$ experiments, the data are $n$ tuples $(v_1, l_1), (v_2, l_2), \ldots, (v_n, l_n)$, where $l_i$ is the observed intensity at voltage $v_i$. The observing device is imprecise, so the data may contain errors. ACM R&D wants the estimated intensity function $F$ that minimizes the error function

$\text{error}(F) = \max_{1 \le i \le n} \left| l_i - F(v_i) \right|$

Write a program that finds such an $F$ and outputs the value of $\text{error}(F)$.

The first line contains an integer $n$ ($1 \le n \le 300{,}000$), the number of tuples. Each of the following $n$ lines contains two integers $v_i$ and $l_i$ ($0 \le v_i, l_i \le 10^9$). No two tuples have the same $v_i$.

Print exactly one line containing one real number, the minimum value of $\text{error}(F)$, rounded to the first decimal place.