A9186. Sereja and Sets

Sereja has $m$ non-empty sets of integers $A_{1},A_{2},...,A_{m}$ . What a lucky coincidence! The given sets are a partition of the set of all integers from 1 to $n$ . In other words, for any integer $v$ $(1<=v<=n)$ there is exactly one set $A_{t}$ such that . Also Sereja has integer $d$ .

Sereja decided to choose some sets from the sets he has. Let's suppose that $i_{1},i_{2},...,i_{k}$ $(1<=i_{1}<i_{2}<...<i_{k}<=m)$ are indexes of the chosen sets. Then let's define an array of integers $b$ , sorted in ascending order, as a union of the chosen sets, that is, . We'll represent the element with number $j$ in this array (in ascending order) as $b_{j}$ . Sereja considers his choice of sets correct, if the following conditions are met:

$b_{1}<=d; b_{i+1}-b_{i}<=d (1<=i<|b|); n-d+1<=b_{|b|}.$ Sereja wants to know what is the minimum number of sets $(k)$ that he can choose so that his choice will be correct. Help him with that.

The first line contains integers $n$ , $m$ , $d$ $(1<=d<=n<=10^{5},1<=m<=20)$ . The next $m$ lines contain sets. The first number in the $i$ -th line is $s_{i}$ $(1<=s_{i}<=n)$ . This number denotes the size of the $i$ -th set. Then the line contains $s_{i}$ distinct integers from 1 to $n$ — set $A_{i}$ .

It is guaranteed that the sets form partition of all integers from 1 to $n$ .

In a single line print the answer to the problem — the minimum value $k$ at the right choice.