
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
Given the determinant of order  $n$   $$ \begin{vmatrix}
   8   &  3   &  3  &  \dots  & 3 
   3   &  8   &  3  &  \dots  & 3 
   3   &  3   &  8  &  \dots   & 3 
\vdots &  \vdots  &  \vdots &  \ddots & \vdots 
   3   &  3   &  3  &  \dots  & 8
\end{vmatrix} $$  Calculate its value and determine for which values of  $n$  this value is a multiple of  $10$ .

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