
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
Given \(152, a_{1}, a_{2}, \cdots\) is an integer sequence satisfying \(a_{1} \cdot a_{2} + a_{2} \cdot a_{3} + \cdots + a_{n-1} \cdot a_{n} = \frac{(n-1) \cdot n \cdot (n+1)}{3}\) for \(n = 2, 3, \cdots\), find the number of such sequences.

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