
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  $p$  be an odd prime number. Let  $g$  be a primitive root of unity modulo  $p$ . Find all the values of  $p$  such that the sets  $A=\left\{k^2+1:1\le k\le\frac{p-1}2\right\}$  and  $B=\left\{g^m:1\le m\le\frac{p-1}2\right\}$  are equal modulo  $p$ .

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