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 \(1 \le t_1 < \ldots < t_k \le 2022\) so that \(t_{i+1} - t_i \le 2\) for all \(i\), and

\[\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.\]