50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)

Consider an arbitrary arrangement of the brackets. Numbers $2010$ and $2009$ always have signs '+' and '-', respectively, that's why the sum reaches the maximum value when the other terms will have '+' sign.

In the expression $$\frac{2010-2009-2010-2009-2010-2009-\cdots -2010-2009}{2010\text{ numbers}}$$ there are somehow placed brackets and the value is calculated. Find the maximum value that can be reached. Justify your answer.

Notice. Left bracket can be placed only before a number and right - only after. For example, expressions $-2010(-2009-2010)$ and $-(2010-2009-)2010$ are incorrect.