CF1714F.Build a Tree and That Is It

A tree is a connected undirected graph without cycles. Note that in this problem, we are talking about not rooted trees.

You are given four positive integers $n, d_{12}, d_{23}$ and $d_{31}$. Construct a tree such that:

• it contains $n$ vertices numbered from $1$ to $n$,
• the distance (length of the shortest path) from vertex $1$ to vertex $2$ is $d_{12}$,
• distance from vertex $2$ to vertex $3$ is $d_{23}$,
• the distance from vertex $3$ to vertex $1$ is $d_{31}$.

Output any tree that satisfies all the requirements above, or determine that no such tree exists.

树是一个无环的连通无向图。注意，在本题中，我们讨论的是非有根树（即不指定根节点的树）。

给定四个正整数 $n, d_{12}, d_{23}$ 和 $d_{31}$。请构造一棵满足以下条件的树：

• 该树包含 $n$ 个顶点，编号为 $1$ 到 $n$；
• 顶点 $1$ 到顶点 $2$ 的距离（即最短路径的长度）为 $d_{12}$；
• 顶点 $2$ 到顶点 $3$ 的距离为 $d_{23}$；
• 顶点 $3$ 到顶点 $1$ 的距离为 $d_{31}$。

输出任意一棵满足上述所有条件的树；若不存在这样的树，则判定其不存在。

The first line of the input contains an integer $t$ ($1 \le t \le 10^4$) —the number of test cases in the test.

This is followed by $t$ test cases, each written on a separate line.

Each test case consists of four positive integers $n, d_{12}, d_{23}$ and $d_{31}$ ($3 \le n \le 2\cdot10^5; 1 \le d_{12}, d_{23}, d_{31} \le n-1$).

It is guaranteed that the sum of $n$ values for all test cases does not exceed $2\cdot10^5$.

输入的第一行包含一个整数 $t$（$1 \le t \le 10^4$），表示测试用例的数量。

接下来是 $t$ 个测试用例，每个测试用例占单独一行。

每个测试用例包含四个正整数 $n, d_{12}, d_{23}$ 和 $d_{31}$（$3 \le n \le 2\cdot10^5$；$1 \le d_{12}, d_{23}, d_{31} \le n-1$）。

保证所有测试用例的 $n$ 值之和不超过 $2\cdot10^5$。

For each test case, print YES if the suitable tree exists, and NO otherwise.

If the answer is positive, print another $n-1$ line each containing a description of an edge of the tree — a pair of positive integers $x_i, y_i$, which means that the $i$th edge connects vertices $x_i$ and $y_i$.

The edges and vertices of the edges can be printed in any order. If there are several suitable trees, output any of them.

对于每个测试用例，如果存在满足条件的树，则输出 YES；否则输出 NO。

如果答案为 YES，则再输出 $n-1$ 行，每行描述树中的一条边——即一对正整数 $x_i, y_i$，表示第 $i$ 条边连接顶点 $x_i$ 和 $y_i$。

边及其顶点的输出顺序可以任意。如果存在多个满足条件的树，输出任意一个即可。