Let $A$ be an unknown matrix of the form: $$ A = \left( \begin{matrix} b & c_1 & c_2 \\ c_1 & c_2 & c_3 \\ c_2 & c_3 & c_4 \\ c_3 & c_4 & d\end{matrix} \right) $$
I have the known vectors $$ x = \left( \begin{matrix} x_1 \\ x_2\\ x_3\end{matrix} \right) $$ and
$$ y = \left( \begin{matrix} y_1 \\ y_2\\ y_3\\y_4\end{matrix} \right) $$
The relation between these is $Ax=y$
Is there a way to solve for $c_1, c_2, c_3, c_4$ knowing $x$ and $y$?
Any hints and help is appreciated!