System of Inequalities in Continued Fractions from Finite Alphabets

The system of two inequalities
$$ \biggl|\frac yx-\psi_1\biggr|\le \varepsilon_1\qquad \text{and} \qquad \biggl\|\frac{ay}x-\psi_2\biggr\|\le \varepsilon_2 $$
is considered, and an upper bound for the number of its solutions is established. Here $a$, $\psi_1$, $\psi_2$, $\varepsilon_1$, and $\varepsilon_2$ are given real numbers, $\varepsilon_1$ and $\varepsilon_1$ are positive and arbitrarily small, $\|\cdot\|$ is the distance to the nearest integer, and $x$ and $y$ are coprime variables from given intervals such that the partial quotients of the continued fraction expansion of $y/x$ belong to a finite alphabet $\mathbf{A}\subseteq\mathbb{N}$.