1. Solve the following system of equations in the domain of real numbers

$$
\begin{aligned}
2 x+\lfloor y\rfloor & =2022, \\
3 y+\lfloor 2 x\rfloor & =2023 .
\end{aligned}
$$

(The symbol $\lfloor a\rfloor$ denotes the lower integer part of a real number a, i.e. the greatest integer not greater than a. E.g. $\lfloor 1,9\rfloor=1$ and $\lfloor-1,1\rfloor=-2$.) (Jaroslav Švrček)