HMMT 2011 U12

A sequence of integers $\left\{a_{i}\right\}$ is defined as follows: $a_{i}=i$ for all $1 \leq i \leq 5$, and $a_{i}=a_{1} a_{2} \cdots a_{i-1}-1$ for all $i>5$. Evaluate $a_{1} a_{2} \cdots a_{2011}-\sum_{i=1}^{2011} a_{i}^{2}$.