
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
Let \( S_{0} = 0 \) and let \( S_{k} = a_{1} + 2a_{2} + \ldots + k a_{k} \) for \( k \geq 1 \). Define \( a_{i} \) to be 1 if \( S_{i-1} < i \) and -1 if \( S_{i-1} \geq i \). What is the largest \( k \leq 2010 \) such that \( S_{k} = 0 \)?

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