
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 a prime number and let  $k$  be a positive integer. Suppose that the numbers  $a_i=i^k+i$  for  $i=0,1, \ldots,p-1$  form a complete residue system modulo  $p$ . What is the set of possible remainders of  $a_2$  upon division by  $p$ ?

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