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" $\to$ "yz", "y" $\to$ "zx", "z" $\to$ "xy";
• exchange two consecutive different letters in the string, e.g. "xy" $\to$ "yx", "zx" $\to$ "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.

We define a sequence $(a_n{)}_{n\ge 0}$: $a_0=88888888$ and $a_n=(8888\cdot a_{n-1})mod888888883$ for $n>0$.

Let $b_n=a_{n}mod3$. For each $i\ge 0$, a string $c(i)$ of length $50$ is defined by translating the finite sequence $b_{50i},b_{50i+1},\dots ,b_{50i+49}$ via the rule: $0\to$ "x", $1\to$ "y", $2\to$ "z".

Let $F(N)$ be the number of ordered pairs $(i,j)$ with $0\le i,j<N$ such that the concatenated string $c(i)c(j)$ is neutral.
For example, $F(10)=13$ and $F(100)=1224$.

Find $F({10}^6)$.