
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
Find the number of positive integers  $n$  for which there exists a sequence  $x_1, x_2, \cdots, x_n$  of integers with the following property: if indices  $1 \le i \le j \le n$  satisfy  $i+j \le n$  and  $x_i - x_j$  is divisible by  $3$ , then  $x_{i+j} + x_i + x_j + 1$  is divisible by  $3$ .

*Based on a proposal by Ivan Koswara*

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