
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  $k > 1, n > 2018$  be positive integers, and let  $n$  be odd. The nonzero rational numbers  $x_1,x_2,\ldots,x_n$  are not all equal and satisfy   $$ x_1+\frac{k}{x_2}=x_2+\frac{k}{x_3}=x_3+\frac{k}{x_4}=\ldots=x_{n-1}+\frac{k}{x_n}=x_n+\frac{k}{x_1} $$  Find:
a)	the product  $x_1 x_2 \ldots x_n$  as a function of  $k$  and  $n$ b)	the least value of  $k$ , such that there exist  $n,x_1,x_2,\ldots,x_n$  satisfying the given conditions.

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