
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]


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Let  $n \ge 2$  be a fixed integer. [list=a] [*]Determine the largest positive integer  $m$  (in terms of  $n$ ) such that there exist complex numbers  $r_1$ ,  $\dots$ ,  $r_n$ , not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of  $m$ , find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list]

*Kaixin Wang*

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