<s>[INST] This problem was on our chrismas math contest and one of the teams solved it in under 3 minutes.

For how many integers $n$ does the inequality $1 \leq a(n) \leq 6$ hold, where the function $a(n)$ is defined as $a(n) = 2(2n + 1)(n + 2)(2n^2 + 3n + 1).$?[/INST]