Problem Statement

Takahashi has a tree $T$. $T$ consists of at most $500$ vertices, and each edge has a positive integer weight.

Using the tree $T$, Takahashi performed the following operations.

1. Choose some of the vertices of the tree $T$ to form a vertex set $S$.
2. For every combination of two distinct vertices chosen from $S$, compute the distance between the two vertices and write each distance on a blackboard. Here, the distance between two vertices is the sum of the weights of the edges on the simple path connecting those two vertices in the tree $T$.

You are given a set $D = \{D_1, D_2, \dots, D_K\}$ consisting of $K$ elements. Each element of $D$ is a positive even number. It is known that every number written on the blackboard is contained in $D$, and that every number contained in $D$ was written on the blackboard at least once.

Find one possible combination of the tree $T$ and the vertex set $S$ created by Takahashi. It can be shown that such a combination of a tree and a vertex set always exists under the constraints of this problem.

Input

$K$
$D_1\ D_2\dots D_K$

• $1 \le K \le 150$
• $2 \le D_i \le 300$
• $D_i$ is even
• If $i \neq j$, then $D_i \neq D_j$
• All inputs are integers

Output

Output one possible combination of the tree $T$ and the vertex set $S$ chosen by Takahashi in the following format.

$N$
$u_1\ v_1\ w_1$
$u_2\ v_2\ w_2$
$\vdots$
$u_{N-1}\ v_{N-1}\ w_{N-1}$
$M$
$S_1\ S_2\dots S_M$

On line $1$, output the number of vertices $N(2 \le N \le 500)$ of the tree $T$.

On each of lines $2$ through $N$, output the information of the $N-1$ edges of the tree $T$, one edge per line. On line $i+1$, output $3$ integers $u_i, v_i, w_i(1 \le u_i, v_i \le N, 1 \le w_i \le 1000)$ separated by spaces. This output means that the $i$-th edge of the tree $T$ bidirectionally connects vertex $u_i$ and vertex $v_i$ with weight $w_i$.

On line $N+1$, output the size $M(2 \le M \le N)$ of the vertex set $S$.

On line $N+2$, output the numbers of the $M$ vertices contained in $S$, separated by spaces. $S_1, S_2 \dots S_M$ must be pairwise distinct integers between $1$ and $N$, inclusive.

Sample

3
6 2 4

4
1 2 2
2 3 2
3 4 2
3
1 2 4