Given two unequal positive integers $a$ and $b$, we define a self-describing sequence consisting of alternating runs of $a$s and $b$s. The first element is $a$ and the sequence of run lengths is the original sequence.

For $a=2,b=3$, the sequence is: $$2,2,3,3,2,2,2,3,3,3,2,2,3,3,2,2,3,3,3,2,2,2,3,3,3,...$$ The sequence begins with two $2$s and two $3$s, then three $2$s and three $3$s, so the run lengths $2,2,3,3,...$ are given by the original sequence.

Let $T(a,b,N)$ be the sum of the first $N$ elements of the sequence. You are given $T(2,3,10)=25$, $T(4,2,{10}^4)=30004$, $T(5,8,{10}^6)=6499871$.

Find $\sum T(a,b,22332223332233)$ for $2\le a\le 223$, $2\le b\le 223$ and $a\ne b$. Give your answer modulo $2233222333$.