
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  $ m$  a positive integer and  $ p$  a prime number, both fixed. Define  $ S$  the set of all  $ m$ -uple of positive integers  $ \vec{v} \equal{} (v_1,v_2,\ldots,v_m)$  such that  $ 1 \le v_i \le p$  for all  $ 1 \le i \le m$ . Define also the function  $ f(\cdot): \mathbb{N}^m \to \mathbb{N}$ , that associates every  $ m$ -upla of non negative integers  $ (a_1,a_2,\ldots,a_m)$  to the integer  $ \displaystyle f(a_1,a_2,\ldots,a_m) \equal{} \sum_{\vec{v} \in S} \left(\prod_{1 \le i \le m}{v_i^{a_i}} \right)$ . 
Find all  $ m$ -uple of non negative integers  $ (a_1,a_2,\ldots,a_m)$  such that  $ p \mid f(a_1,a_2,\ldots,a_m)$ .
*(Pierfrancesco Carlucci)*

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