
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 with unknown and potentially different gas pressures. It is permitted to connect any number of cylinders together, not exceeding a given natural number $k$, and then disconnect them; during the connection, the gas pressure in the connected cylinders becomes equal to the arithmetic mean of their pressures before the connection. What is the smallest $k$ for which there exists a way to equalize the pressures in all 40 cylinders, regardless of the initial distribution of pressures in the cylinders?

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