Find all the values of $a$ for which the equations
\begin{align*}
x^2 + ax + 1 &= 0, \\
x^2  - x - a &= 0
\end{align*}have a common real root.  Enter all the possible values, separated by commas.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find all the values of $a$ for which the equations
\begin{align*}
x^2 + ax + 1 &= 0, \\
x^2  - x - a &= 0
\end{align*}have a common real root.  Enter all the possible values, separated by commas.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Find all the values of $a$ for which the equations
\begin{align*}
x^2 + ax + 1 &= 0, \\
x^2  - x - a &= 0
\end{align*}have a common real root.  Enter all the possible values, separated by commas.
train