<s>[INST] Find the possible value from the following. Find the possible value from the following.
I'm not able to end up on a concrete note, as I'm unable to get the essence of question, still not clear to me. 

$x$, $y$, $z$ are distinct reals such that $y=x(4-x)$, $z=y(4-y)$, $x=z(4-z)$. The possible values of $x+y+z$ is:
$$\begin{array}{l} A.\ 4 && C.\ 7 \\ B.\ 6 && D.\ 9 \end{array}$$

 [/INST] Composing the functions, we get
$$
\begin{align}
0
&=x^8-16x^7+104x^6-352x^5+660x^4-672x^3+336x^2-63x\\
&=x(x-3)(x^3-7x^2+14x-7)(x^3-6x^2+9x-3)\tag{1}
\end{align}
$$
The roots $x=0$ and $x=3$ lead to indistinct $x$, $y$, and $z$.
$x^3-7x^2+14x-7$ has 3 real roots in $[0,4]$ whose sum is $7$ (the negative of the coefficient of $x^2$).
$x^3-6x^2+9x-3$ has 3 real roots in $[0,4]$ whose sum is $6$ (the negative of the coefficient of $x^2$).
$x$, $y$, and $z$ all satisfy $(1)$.
$t(4-t)$ rotates the roots of the cubics.
Thus, the possible values of $x+y+z$ are $6$ and $7$.
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