
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. Discuss the number of solutions for \(x\) in the equation \( |x+1| + |x+2| + |x+3| = a \).

2. Consider the arithmetic sequence \(a_1, a_2, \ldots, a_n\) for which
   \[
   \left| a_1 \right| + \left| a_2 \right| + \cdots + \left| a_n \right| = 
   \left| a_1 + 1 \right| + \left| a_2 + 1 \right| + \cdots + \left| a_n + 1 \right| = 
   \left| a_1 - 2 \right| + \left| a_2 - 2 \right| + \cdots + \left| a_n - 2 \right| = 507,
   \]
   determine the maximum number of terms \( n \).

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