Railway

A couple of years ago the Bergen Ministry of Infrastructure prepared a plan for a new light railway network. This network was supposed to connect all $n$ neighbourhoods in the city with $n - 1$ railway tracks in such a way, that there would be a path from every neighbourhood to every other neighbourhood. The planned tracks are identified by numbers from $1$ to $n - 1$.

Years passed, new elections are approaching, and the railway
network still exists only on paper. Therefore the Minister of
Infrastructure (representing a party holding disagreement in
high regard) decided to construct at least some part of the
plan. He asked each of his $m$ deputy ministers to choose which
neighbourhoods *they* thought should be connected. That
will result in a list of necessary tracks for each deputy
minister. If a deputy minister thinks that the neighbourhoods
$a_1, \ldots , a_ s$ need
to be connected, then according to him or her, the necessary
tracks are all those which lie on planned paths from
$a_ i$ to $a_ j$ for some $1 \leq i < j \leq s$.

The minister just received all lists from the deputy ministers. He decided to construct in the first place the tracks which are requested by at least $k$ deputy ministers. Your task is to prepare a list of these tracks.

In the first line of the input there are three integers $n$, $m$ and $k$. The next $n - 1$ lines contain the plan; in the $i$-th of these lines there are two integers $a_ i$ and $b_ i$ ($1 \leq a_ i, b_ i \leq n, a_ i \neq b_ i$), specifying that the $i$-th track on the plan is between neighbourhoods $a_ i$ and $b_ i$.

In the next $m$ lines there are neighbourhoods chosen by deputy ministers; the $i$-th of these lines begins with an integer $s_ i$ which specify the number of neighbourhoods chosen by the $i$-th deputy minister. After it there are $s_ i$ integers specifying these neighbourhoods. The total length of all lists of deputy ministers is at most $S$, i.e. $\sum _{i=1}^{m}s_ i \leq S$.

We always have $2 \leq s_ i \leq n \leq 100\, 000$, $S \leq 150\, 000$, and $1 \leq k \leq m \leq 50\, 000$.

In the first line of the output you should write one integer $r$, specifying the number of tracks which are requested by at least $k$ deputy ministers. In the second line you should write $r$ identifiers of these tracks in ascending order.

The first deputy minister thinks that tracks $1$–$3$, $2$–$3$, $3$–$4$ and $4$–$5$ are necessary. The second deputy minister considers tracks $3$–$4$ and $4$–$6$, and the third one only track $2$–$3$. Tracks $2$–$3$ and $3$–$4$ are necessary according to at least two deputy ministers.

Sample Input 1 | Sample Output 1 |
---|---|

6 3 2 1 3 2 3 3 4 6 4 4 5 4 1 3 2 5 2 6 3 2 3 2 |
2 2 3 |