smc

Find the LCMs of the groups of the numbers. Notice that the groups are relatively prime. So $a_n=$: 11 if $n$ is a multiple of 182. 13 if $n$ is a multiple of 154. 14 if $n$ is a multiple of 143. When do we see ambiguities (for example: $n$ is a multiple of 11, 13, and 14)? This is only done when $n$ is a multiple of $\mathrm{lcm}(11,13,14)=2002$. However, since $n<2002$, this can never happen. So we have 10 multiples of 182 we have to count (1 to 10 $\ast 182$), and similarly, 12 multiples of 154, and 13 multiples of 143. The sum is $10\ast 11+12\ast 13+13\ast 14=448$. Select $\boxed{448}$.