
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 some fixed positive integer  $n>2$ , suppose  $x_1$ ,  $x_2$ ,  $x_3$ ,  $\ldots$  is a nonconstant sequence of real numbers such that  $x_i=x_j$  if  $i \equiv j \pmod{n}$ . Let  $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$ . Given that    $$ f(1)=f(2)=f(3)=\cdots $$   find all possible values of the product  $x_1 x_2 \ldots x_n$ .

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