Question 104
Suppose that 6 real numbers $a, b, c, d, e, f$ satisfy the following equations:
$$
\begin{aligned}
& (a + 1999)(b - 1999)(c + 1999) + (d - 1999)(e + 1999)(f - 1999) = 1, \\
& (a + 2000)(b - 2000)(c + 2000) + (d - 2000)(e + 2000)(f - 2000) = 10, \\
& (a + 2001)(b - 2001)(c + 2001) + (d - 2001)(e + 2001)(f - 2001) = 100.
\end{aligned}
$$
Determine the value of the following quantity:
$$
(a + 2009)(b - 2009)(c + 2009) + (d - 2009)(e + 2009)(f - 2009).
$$
$$
\begin{aligned}
& (a + 1999)(b - 1999)(c + 1999) + (d - 1999)(e + 1999)(f - 1999) = 1, \\
& (a + 2000)(b - 2000)(c + 2000) + (d - 2000)(e + 2000)(f - 2000) = 10, \\
& (a + 2001)(b - 2001)(c + 2001) + (d - 2001)(e + 2001)(f - 2001) = 100.
\end{aligned}
$$
Determine the value of the following quantity:
$$
(a + 2009)(b - 2009)(c + 2009) + (d - 2009)(e + 2009)(f - 2009).
$$