
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
Let  $S_{n}=\{1,n,n^{2},n^{3}, \cdots \}$ , where  $n$  is an integer greater than  $1$ . Find the smallest number  $k=k(n)$  such that there is a number which may be expressed as a sum of  $k$  (possibly repeated) elements in  $S_{n}$  in more than one way. (Rearrangements are considered the same.)

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