$$
\begin{array}{l}
a^{2}+b^{2}+c^{2}+d^{2}+(1-a)^{2}+(a-b)^{2}+ \\
(b-c)^{2}+(c-d)^{2}+(d-1)^{2}
\end{array}
$$

When the value of the above algebraic expression is minimized, what is the value of $a+b+c+d$? Prove your conclusion.