Continued fraction/Arithmetic/G(matrix ng, continued fraction n1, continued fraction n2)

This task performs the basic mathematical functions on 2 continued fractions. This requires the full version of matrix NG:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$

I may perform perform the following operations:

Input the next term of continued fraction N₁

Input the next term of continued fraction N₂

Output a term of the continued fraction resulting from the operation.

I output a term if the integer parts of ${\frac {a}{b}}$ and ${\frac {a_{1}}{b_{1}}}$ and ${\frac {a_{2}}{b_{2}}}$ and ${\frac {a_{12}}{b_{12}}}$ are equal. Otherwise I input a term from continued fraction N₁ or continued fraction N₂. If I need a term from N but N has no more terms I inject $\infty$.

When I input a term t from continued fraction N₁ I change my internal state:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a_{2}+a_{12}*t&a+a_{1}*t&a_{12}&a_{1}\\b_{2}+b_{12}*t&b+b_{1}*t&b_{12}&b_{1}\end{bmatrix}}$

When I need a term from exhausted continued fraction N₁ I change my internal state:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a_{12}&a_{1}&a_{12}&a_{1}\\b_{12}&b_{1}&b_{12}&b_{1}\end{bmatrix}}$

When I input a term t from continued fraction N₂ I change my internal state:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a_{1}+a_{12}*t&a_{12}&a+a_{2}*t&a_{2}\\b_{1}+b_{12}*t&b_{12}&b+b_{2}*t&b_{2}\end{bmatrix}}$

When I need a term from exhausted continued fraction N₂ I change my internal state:

${\begin{bmatrix}a_{12}&a_{2}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}a_{12}&a_{12}&a_{2}&a_{2}\\b_{12}&b_{12}&b_{2}&b_{2}\end{bmatrix}}$

When I output a term t I change my internal state:

${\begin{bmatrix}a_{12}&a_{1}&a_{2}&a\\b_{12}&b_{1}&b_{2}&b\end{bmatrix}}$ is transposed thus ${\begin{bmatrix}b_{12}&b_{1}&b_{2}&b\\a_{12}-b_{12}*t&a_{1}-b_{1}*t&a_{2}-b_{2}*t&a-b*t\end{bmatrix}}$