Middle European Mathematical Olympiad 2019 Problem T-4

Prove that every integer from $1$ to $2019$ can be represented as an arithmetic expression consisting of up to $17$ symbols $2$ and an arbitrary number of additions, subtractions, multiplications, divisions and brackets. The $2$'s may not be used for any other operation, for example to form multi-digit numbers (such as $222$) or powers (such as $2^2$).

Valid examples: $$\left((2 \times 2 + 2) \times 2 - \frac{2}{2}\right) \times 2 = 22, \quad (2 \times 2 \times 2 - 2) \times \left(2 \times 2 + \frac{2 + 2 + 2}{2}\right) = 42.$$