
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
Find the smallest positive integer $n$ such that there exists a sequence of $n+1$ terms $a_{0}, a_{1}, \cdots, a_{n}$ satisfying $a_{0}=0, a_{n}=2008$, and $\left|a_{i}-a_{i-1}\right|=i^{2}$ for $i=1,2, \cdots, n$.

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