
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
Let \( x_{1}, x_{2}, \ldots, x_{2022} \) be nonzero real numbers. Suppose that \( x_{k} + \frac{1}{x_{k+1}} < 0 \) for each \( 1 \leq k \leq 2022 \), where \( x_{2023} = x_{1} \). Compute the maximum possible number of integers \( 1 \leq n \leq 2022 \) such that \( x_{n} > 0 \).

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