
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  $S_k$  equal  $a_1+2a_2+\ldots+ka_k$  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