
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
There are 40 identical gas cylinders, the pressure values of which are unknown and may vary. It is allowed to connect any cylinders with each other in a number not exceeding a given natural number \( k \), and then disconnect them; during which the gas pressure in the connected cylinders is set to the arithmetic mean of the pressures in them before the connection. What is the smallest \( k \) for which there exists a method to equalize the pressures in all 40 cylinders regardless of their initial pressure distribution?

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