
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
In a regular $(n+2)$-gon, two vertices are randomly selected and a line passing through them is drawn. Let $p_{n}$ denote the probability that the difference in the number of vertices on the two sides of this line does not exceed 1. Calculate the value of $\sum_{i=1}^{2018} p_{i} p_{i+1}$.

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