
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
Given an integer \( n > 1 \), there are \( n^2 \) cable car stations on a mountain, each at a different height. Two cable car companies, \( A \) and \( B \), each operate \( k \) cable cars. Each cable car runs from one station to a higher station (without stops at intermediate stations). For company \( A \), the \( k \) starting points and \( k \) ending points are all distinct, and a higher starting point corresponds to a higher ending point. The cable cars of company \( B \) satisfy the same conditions. A station is said to be "connected" by a company if it is possible to travel from the lower station to the higher station using one or more of the company's cable cars (without any other movement between stations). Find the smallest positive integer \( k \) such that there must be two stations that are simultaneously connected by both companies.

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