
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 \(1 \leq t_1 < \ldots < t_k \leq 2022\) such that \(t_{i+1} - t_i \leq 2\) for all \(i\), and
$$
\left|\sum_{i=1}^{k} a_{t_i}\right| \geq C.
$$

Present the answer in LaTex format: \boxed{Your answer}
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