
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
On Lineland there are 2018 bus stations numbered 1 through 2018 from left to right. A self-driving bus that can carry at most  $N$  passengers starts from station 1 and drives all the way to station 2018, while making a stop at each bus station. Each passenger that gets on the bus at station  $i$  will get off at station  $j$  for some  $j>i$  (the value of  $j$  may vary over different passengers). Call any group of four distinct stations  $i_1, i_2, j_1, j_2$  with  $i_u< j_v$  for all  $u,v\in \{1,2\}$  a *good* group. Suppose that in any good group  $i_1, i_2, j_1, j_2$ , there is a passenger who boards at station  $i_1$  and de-boards at station  $j_1$ , or there is a passenger who boards at station  $i_2$  and de-boards at station  $j_2$ , or both scenarios occur. Compute the minimum possible value of  $N$ .

*Proposed by Yannick Yao*

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