Anna, Svyatoslav and Maps

Description:

Input:

The first line contains a single integer $$$n$$$ ($$$2 \le n \le 100$$$) — the number of vertexes in a graph.

The next $$$n$$$ lines define the graph by an adjacency matrix: the $$$j$$$-th character in the $$$i$$$-st line is equal to $$$1$$$ if there is an arc from vertex $$$i$$$ to the vertex $$$j$$$ else it is equal to $$$0$$$. It is guaranteed that the graph doesn't contain loops.

The next line contains a single integer $$$m$$$ ($$$2 \le m \le 10^6$$$) — the number of vertexes in the path.

The next line contains $$$m$$$ integers $$$p_1, p_2, \ldots, p_m$$$ ($$$1 \le p_i \le n$$$) — the sequence of vertexes in the path. It is guaranteed that for any $$$1 \leq i < m$$$ there is an arc from $$$p_i$$$ to $$$p_{i+1}$$$.

Output:

In the first line output a single integer $$$k$$$ ($$$2 \leq k \leq m$$$) — the length of the shortest good subsequence. In the second line output $$$k$$$ integers $$$v_1$$$, $$$\ldots$$$, $$$v_k$$$ ($$$1 \leq v_i \leq n$$$) — the vertexes in the subsequence. If there are multiple shortest subsequences, print any. Any two consecutive numbers should be distinct.

Sample Input:

4
0110
0010
0001
1000
4
1 2 3 4

Sample Output:

3
1 2 4 

Sample Input:

4
0110
0010
1001
1000
20
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Sample Output:

11
1 2 4 2 4 2 4 2 4 2 4 

Sample Input:

3
011
101
110
7
1 2 3 1 3 2 1

Sample Output:

7
1 2 3 1 3 2 1 

Sample Input:

4
0110
0001
0001
1000
3
1 2 4

Sample Output:

2
1 4 

Note:

Below you can see the graph from the first example:

The given path is passing through vertexes $$$1$$$, $$$2$$$, $$$3$$$, $$$4$$$. The sequence $$$1-2-4$$$ is good because it is the subsequence of the given path, its first and the last elements are equal to the first and the last elements of the given path respectively, and the shortest path passing through vertexes $$$1$$$, $$$2$$$ and $$$4$$$ in that order is $$$1-2-3-4$$$. Note that subsequences $$$1-4$$$ and $$$1-3-4$$$ aren't good because in both cases the shortest path passing through the vertexes of these sequences is $$$1-3-4$$$.

In the third example, the graph is full so any sequence of vertexes in which any two consecutive elements are distinct defines a path consisting of the same number of vertexes.

In the fourth example, the paths $$$1-2-4$$$ and $$$1-3-4$$$ are the shortest paths passing through the vertexes $$$1$$$ and $$$4$$$.