
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  $\mathbb{F}_p$  denote the field of integers modulo a prime  $p,$  and let  $n$  be a positive integer. Let  $v$  be a fixed vector in  $\mathbb{F}_p^n,$  let  $M$  be an  $n\times n$  matrix with entries in  $\mathbb{F}_p,$  and define  $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$  by  $G(x)=v+Mx.$  Let  $G^{(k)}$  denote the  $k$ -fold composition of  $G$  with itself, that is,  $G^{(1)}(x)=G(x)$  and  $G^{(k+1)}(x)=G(G^{(k)}(x)).$  Determine all pairs  $p,n$  for which there exist  $v$  and  $M$  such that the  $p^n$  vectors  $G^{(k)}(0),$   $k=1,2,\dots,p^n$  are distinct.

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