ISL 2022 C1

A $\pm 1$-sequence is a sequence of 2022 numbers $a_{1}, \ldots, a_{2022}$, each equal to either +1 or -1 . Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \leqslant t_{1}<\ldots<t_{k} \leqslant 2022$ so that $t_{i+1}-t_{i} \leqslant 2$ for all $i$, and $$\left|\sum_{i=1}^{k} a_{t_{i}}\right| \geqslant C$$ (Czech Republic)