<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Let  $p = 2017$ . If  $A$  is an  $n\times n$  matrix composed of residues  $\pmod{p}$  such that  $\det A\not\equiv 0\pmod{p}$  then let  $\text{ord}(A)$  be the minimum integer  $d > 0$  such that  $A^d\equiv I\pmod{p}$ , where  $I$  is the  $n\times n$  identity matrix. Let the maximum such order be  $a_n$  for every positive integer  $n$ . Compute the sum of the digits when  $\sum_{k = 1}^{p + 1} a_k$  is expressed in base  $p$ . Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
2015
Let  $p = 2017$ . If  $A$  is an  $n\times n$  matrix composed of residues  $\pmod{p}$  such that  $\det A\not\equiv 0\pmod{p}$  then let  $\text{ord}(A)$  be the minimum integer  $d > 0$  such that  $A^d\equiv I\pmod{p}$ , where  $I$  is the  $n\times n$  identity matrix. Let the maximum such order be  $a_n$  for every positive integer  $n$ . Compute the sum of the digits when  $\sum_{k = 1}^{p + 1} a_k$  is expressed in base  $p$ .
test