
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
(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  $n$ .

Present the answer in LaTex format: \boxed{Your answer}
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