Let $a$ be a real number in the open interval $]0,1[$. Show that there exists $\lambda > 0$ such that the polynomial
$$P(x) = x - \lambda x(x-a)(x-1)$$
satisfies the following two properties:
\begin{enumerate}
\item $P([0,1]) = [0,1]$,
\item $P$ is increasing on $[0,1]$.
\end{enumerate}