981. The Quaternion Group II

Starting from an empty string, we want to build a string with letters "x", "y", "z". At each step, one of the following operations is performed:

• insert two consecutive identical letters "xx", "yy" or "zz" anywhere into the string;

• replace one letter in the string with two consecutive letters, according to the rule: "x" $\to$$\to$ "yz", "y" $\to$$\to$ "zx", "z" $\to$$\to$ "xy";

• exchange two consecutive different letters in the string, e.g. "xy" $\to$$\to$ "yx", "zx" $\to$$\to$ "xz", etc.

A string is called neutral if it is possible to produce the string from the empty string after an even number of steps.

Let $N(X,Y,Z)$$N(X, Y, Z)$ be the number of neutral strings which contain $X$$X$ copies of "x", $Y$$Y$ copies of "y" and $Z$$Z$ copies of "z".
For example, $N(2,2,2)=42$$N(2, 2, 2) = 42$ and $N(8,8,8)=4732773210$$N(8, 8, 8) = 4732773210$.

Find the sum of $N(i^3,j^3,k^3)$$N(i^3, j^3, k^3)$ for $0\le i,j,k<88$$0 \le i, j, k \lt 88$. Give your answer modulo $888888883$$888\,888\,883$.

981. 四元群 2

从一个空串开始，我们希望构造一个仅含字母 'x'、'y'、'z' 的字符串。构造的每一步中，你只能进行如下操作之一：

• 在这个字符串中的任意处插入两个连续且相同的字符 "xx"、"yy" 或 "zz"。

• 依照如下规则，把字符串中的一个字符替换成两个连续字符："x" $\to$$\to$ "yz"、"y" $\to$$\to$ "zx"、"z" $\to$$\to$ "xy"。

• 交换字符串中两个连续且 不同 的字符。例如："xy" $\to$$\to$ "yx"、"zx" $\to$$\to$ "xz" 等。

若能够从空串开始，经 偶数 次操作后得到某个字符串，则称这个字符串是 中性的。

记 $N(X,Y,Z)$$N(X, Y, Z)$ 为含 $X$$X$ 个 "x"、$Y$$Y$ 个 "y"、$Z$$Z$ 个 "z" 的中性的字符串的数量。
例如 $N(2,2,2)=42$$N(2, 2, 2) = 42$、$N(8,8,8)=4732773210$$N(8, 8, 8) = 4732773210$。

求 $N(i^3,j^3,k^3)$$N(i^3, j^3, k^3)$ 的和模 $888888883$$888\,888\,883$ 的值，其中 $0\le i,j,k<88$$0 \le i, j, k \lt 88$。

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