
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 an integer  $n\ge\ 3$ , find the least positive integer  $k$ , such that there exists a set  $A$  with  $k$  elements, and  $n$  distinct reals  $x_{1},x_{2},\ldots,x_{n}$  such that  $x_{1}+x_{2}, x_{2}+x_{3},\ldots, x_{n-1}+x_{n}, x_{n}+x_{1}$  all belong to  $A$ .

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