
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  $1 \le t_1 < \ldots < t_k \le 2022$  so that  $t_{i+1} - t_i \le 2$  for all  $i$ , and   $$ \left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C. $$  

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