CN-SMO 2005 Problem 7

(1) Find the possible number of roots for the equation $|x + 1| + |x + 2| + |x + 3| = a$ , where $x \in R$ and $a$ is parameter.

(2) Let $\{ a_1, a_2, \ldots, a_n \}$ be an arithmetic progression, $n \in \mathbb{N}$ , and satisfy the condition $$\sum^{n}_{i=1}|a_i| = \sum^{n}_{i=1}|a_{i} + 1| = \sum^{n}_{i=1}|a_{i} - 2| = 507.$$ Find the maximum value of $n$ .