CF451C.Predict Outcome of the Game

There are n games in a football tournament. Three teams are participating in it. Currently k games had already been played.

You are an avid football fan, but recently you missed the whole k games. Fortunately, you remember a guess of your friend for these k games. Your friend did not tell exact number of wins of each team, instead he thought that absolute difference between number of wins of first and second team will be _d_1 and that of between second and third team will be _d_2.

You don't want any of team win the tournament, that is each team should have the same number of wins after n games. That's why you want to know: does there exist a valid tournament satisfying the friend's guess such that no team will win this tournament?

Note that outcome of a match can not be a draw, it has to be either win or loss.

一场足球锦标赛共有 $n$ 场比赛，有三支队伍参加。目前已有 $k$ 场比赛结束。

你是一位狂热的足球迷，但最近你错过了全部这 $k$ 场比赛。幸运的是，你还记得朋友对这 $k$ 场比赛结果所作的一个猜测：朋友并未告诉你每支队伍确切的胜场数，而是猜测：第一支队伍与第二支队伍的胜场数之差的绝对值为 $d_1$，第二支队伍与第三支队伍的胜场数之差的绝对值为 $d_2$。

你不希望任何一支队伍最终赢得整个锦标赛，即：在全部 $n$ 场比赛结束后，三支队伍的胜场数必须完全相等。因此你想知道：是否存在一种符合朋友猜测的、合法的比赛结果安排，使得最终没有任何一支队伍获胜？

注意：每场比赛的结果不能为平局，只能是一方获胜、另一方失利。

The first line of the input contains a single integer corresponding to number of test cases t (1 ≤ t ≤ 105).

Each of the next t lines will contain four space-separated integers n, k, _d_1, _d_2 (1 ≤ n ≤ 1012; 0 ≤ k ≤ n; 0 ≤ _d_1, _d_2 ≤ k) — data for the current test case.

输入的第一行包含一个整数，表示测试用例的数量 $t$（$1 \leq t \leq 10^5$）。

接下来的 $t$ 行中，每行包含四个以空格分隔的整数 $n$、$k$、$d_1$、$d_2$（$1 \leq n \leq 10^{12}$；$0 \leq k \leq n$；$0 \leq d_1, d_2 \leq k$），表示当前测试用例的数据。

For each test case, output a single line containing either "yes" if it is possible to have no winner of tournament, or "no" otherwise (without quotes).

对于每个测试用例，输出一行，如果有可能使锦标赛没有获胜者，则输出“yes”（不带引号），否则输出“no”（不带引号）。

Sample 1. There has not been any match up to now (k = 0, _d_1 = 0, _d_2 = 0). If there will be three matches (1-2, 2-3, 3-1) and each team wins once, then at the end each team will have 1 win.

Sample 2. You missed all the games (k = 3). As _d_1 = 0 and _d_2 = 0, and there is a way to play three games with no winner of tournament (described in the previous sample), the answer is "yes".

Sample 3. You missed 4 matches, and _d_1 = 1, _d_2 = 0. These four matches can be: 1-2 (win 2), 1-3 (win 3), 1-2 (win 1), 1-3 (win 1). Currently the first team has 2 wins, the second team has 1 win, the third team has 1 win. Two remaining matches can be: 1-2 (win 2), 1-3 (win 3). In the end all the teams have equal number of wins (2 wins).

样例 1：截至目前尚未进行任何比赛（$k = 0$，$d_1 = 0$，$d_2 = 0$）。若接下来进行三场比赛（1–2、2–3、3–1），且每支队伍各赢一场，则最终每支队伍均恰好有 1 场胜利。

样例 2：你错过了全部三场比赛（$k = 3$）。由于 $d_1 = 0$ 且 $d_2 = 0$，而存在一种进行三场比赛且不产生锦标赛冠军的方式（如前一样例所述），因此答案为“是”。

样例 3：你错过了 4 场比赛，且 $d_1 = 1$，$d_2 = 0$。这四场比赛可以是：1–2（2 获胜）、1–3（3 获胜）、1–2（1 获胜）、1–3（1 获胜）。此时第一支队伍有 2 场胜利，第二支队伍有 1 场胜利，第三支队伍有 1 场胜利。剩余两场比赛可以是：1–2（2 获胜）、1–3（3 获胜）。最终，所有队伍的获胜场数相等（均为 2 场胜利）。