Find all real values of $a$ for which the polynomial
\[x^4 + ax^3 - x^2 + ax + 1 = 0\]has at least one real root.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all real values of $a$ for which the polynomial
\[x^4 + ax^3 - x^2 + ax + 1 = 0\]has at least one real root.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\left( -\infty, -\frac{1}{2} \right] \cup \left[ \frac{1}{2}, \infty \right)
Find all real values of $a$ for which the polynomial
\[x^4 + ax^3 - x^2 + ax + 1 = 0\]has at least one real root.
train