
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. Determine the number of tuples $\left(a_{1}, a_{2}, \ldots, a_{p}\right)$ of natural numbers with the following properties:
1) $1 \leq a_{i} \leq p$ for all $i=1, \ldots, p$.
2) $a_{1}+a_{2}+\cdots+a_{p}$ is not divisible by $p$.
3) $a_{1} a_{2}+a_{2} a_{3}+\ldots a_{p-1} a_{p}+a_{p} a_{1}$ is divisible by $p$.

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