
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
For an  $n$ -tuple of integers, define a transformation to be:  $$ (a_1,a_2,\cdots,a_{n-1},a_n)\rightarrow (a_1+a_2, a_2+a_3, \cdots, a_{n-1}+a_n, a_n+a_1) $$  Find all ordered pairs of integers  $(n,k)$  with  $n,k\geq 2$ , such that for any  $n$ -tuple of integers  $(a_1,a_2,\cdots,a_{n-1},a_n)$ , after a finite number of transformations, every element in the of the  $n$ -tuple is a multiple of  $k$ .

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