solution: Note that $x = -1$ is always a root of $x^3 + ax^2 + ax + 1 = 0,$ so we can factor out $x + 1,$ to get
  \[(x + 1) (x^2 + (a - 1) x + 1) = 0.\]The quadratic factor has real roots if and only if its discriminant is nonnegative:
  \[(a - 1)^2 - 4 \ge 0.\]This reduces to $a^2 - 2a - 3 \ge 0,$ which factors as $(a + 1)(a - 3) \ge 0.$  The smallest positive value that satisfies this inequality is $\boxed{3}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $a$ be a positive real number such that all the roots of
  \[x^3 + ax^2 + ax + 1 = 0\]are real.  Find the smallest possible value of $a.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 3
extra_info.question: Let $a$ be a positive real number such that all the roots of
  \[x^3 + ax^2 + ax + 1 = 0\]are real.  Find the smallest possible value of $a.$
extra_info.split: test