
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 a positive integer  $n,$  what is the largest  $k$  such that the numbers  $1,2,\dots,n$  can be put into  $k$  boxes so that the sum of the numbers in each box is the same?

[When  $n=8,$  the example  $\{1,2,3,6\},\{4,8\},\{5,7\}$  shows that the largest  $k$  is *at least* 3.]

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