Line Fitting Under Uncertainty

Fitting a function to a finite set of points sampled from an unknown function $F : \mathbb{R} \to \mathbb{R}$ is a basic problem in mathematics. The most common form is to find a linear function $\overline{F}$ that fits the sampled points best. Suppose $F$ is sampled at $x_1 < x_2 < \dots < x_n$. One way to measure how well $\overline{F}$ fits the sample points is the error

$\text{error}(F, \overline{F}) = \max_{1 \le i \le n} \left| F(x_i) - \overline{F}(x_i) \right|$

The function values at the sample points are not known exactly. Instead you have a discrete probability distribution for each $F(x_i)$: a list of possible values $y_{i,1}, \dots, y_{i,m_i}$ with probabilities $p_{i,j}$ such that $\Pr[F(x_i) = y_{i,j}] = p_{i,j}$. The error is then defined with expected values.

$\text{error}(F, \overline{F}) = \max_{1 \le i \le n} E\left[ \left| F(x_i) - \overline{F}(x_i) \right| \right]$

Write a program that finds the linear function $\overline{F}(x) = ax + b$ minimizing this error and reports the minimum error.

The input holds several test cases. The first line of each test case contains $n$, the number of sample points ($1 \le n \le 10^5$). The next $n$ lines describe the sample points in increasing order of $x$, one line per point. The line for the $i$-th sample point starts with the position $x_i$ at which the function is sampled and the size $m_i$ of its distribution ($0 \le x_i \le 10^9$, $1 \le m_i \le 10$). Then come the $m_i$ possible function values at $x_i$, $y_{i,1}, \dots, y_{i,m_i}$ ($0 \le y_{i,j} < 10^9$), followed by the $m_i$ probabilities $p_{i,1}, \dots, p_{i,m_i}$ ($0 \le p_{i,j} \le 100$). The real probability is $p_{i,j}$ divided by 100, and for each sample point the $m_i$ probability numbers add up to 100. All values are integers and $x_1 < x_2 < \dots < x_n$. The last line of the input contains a single 0, which is not a test case.

For each test case, print one line with the minimum error, rounded to exactly one digit after the decimal point. Round a value exactly halfway up, so 0.25 prints as 0.3.