## Task 2.

A frog is located at the origin $O$ of a number line and needs to perform 2022 jumps, each of a different length from $1, 2, \ldots, 2022$. The jumps must be performed in such an order that the following rules are respected:

- if the frog is currently at point $O$ or to the left of it, it must jump to the right,
- if the frog is to the right of point $O$, it must jump to the left.

Determine the largest natural number $k$ for which the frog can jump in such a way that it never lands on any of the numbers $1, 2, \ldots, k$.