Question:
Let $z$ be a complex number such that
\[z^5 + z^4 + 2z^3 + z^2 + z = 0.\]Find all possible values of $|z|.$

Enter all possible values, separated by commas.

Answer:
First, we can take out a factor of $z,$ to get
\[z(z^4 + z^3 + 2z^2 + z + 1) = 0.\]We can write $z^4 + z^3 + 2z^2 + z + 1 = 0$ as
\[(z^4 + z^3 + z^2) + (z^2 + z + 1) = z^2 (z^2 + z + 1) + (z^2 + z + 1) = (z^2 + 1)(z^2 + z + 1) = 0.\]If $z = 0,$ then $|z| = 0.$

If $z^2 + 1 = 0,$ then $z^2 = -1.$  Taking the absolute value of both sides, we get $|z^2| = 1.$  Then
\[|z|^2 = 1,\]so $|z| = 1.$  (Also, the roots of $z^2 + 1 = 0$ are $z = \pm i,$ both of which have absolute value 1.)

If $z^2 + z + 1 = 0,$ then $(z - 1)(z^2 + z + 1) = 0,$ which expands as $z^3 - 1 = 0.$  Then $z^3 = 1.$  Taking the absolute value of both sides, we get
\[|z^3| = 1,\]so $|z|^3 = 1.$  Hence, $|z| = 1.$

Therefore, the possible values of $|z|$ are $\boxed{0,1}.$