Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\dots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}

Let \(S_0=0\) and let \(S_k\) equal \(a_1+2a_2+\dots +ka_k\) 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  \(S_k=0\)?