
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
We are given  $2001$  balloons and a positive integer  $k$ . Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most  $k$  balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of  $k$  such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.

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