
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 fixed odd prime. A  $p$ -tuple  $(a_1,a_2,a_3,\ldots,a_p)$  of integers is said to be *good* if 



- **(i)**  $0\le a_i\le p-1$  for all  $i$ , and
- **(ii)**  $a_1+a_2+a_3+\cdots+a_p$  is not divisible by  $p$ , and
- **(iii)**  $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$  is divisible by  $p$ .


Determine the number of good  $p$ -tuples.

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