A1023.Robotic Cow Herd--Platinum

Bessie is hoping to fool Farmer John by building a herd of $K$ realistic
robotic cows ($1 \leq K \leq 100,000$).
It turns out that building a robotic cow is somewhat complicated. There are
$N$ ($1 \leq n \leq 100,000$) individual locations on the robot into which
microcontrollers must be connected (so a single microcontroller must be
connected at each location). For each of these locations, Bessie can select
from a number of different models of microcontroller, each varying in cost.
For the herd of robotic cows to look convincing to Farmer John, no two robots
should behave identically. Therefore, no two robots should have exactly the
same set of microcontrollers. For any pair of robots, there should be at least
one location at which the two robots use a different microcontroller model. It
is guaranteed that there will always be enough different microcontroller
models to satisfy this constraint.
Bessie wants to make her robotic herd as cheaply as possible. Help her
determine the minimum possible cost to do this!

The first line of input contains $N$ and $K$ separated by a space.
The following $N$ lines contain a description of the different microcontroller
models available for each location. The $i$th such line starts with $M_i$ ($1 \leq M_i \leq 10$), giving the number of models available for location $i$.
This is followed by $M_i$ space separated integers $P_{i,j}$ giving the costs
of these different models ($1 \le P_{i,j} \le 100,000,000$).

Output a single line, giving the minimum cost to construct $K$ robots.