Induction: Sum of the squares of 6 consecutive natural numbers

Define for every natural n:

$$ a_{n}=\sum\limits_{i=0}^{5}(n+i)^2$$

in other words, $\ a_n$ is the sum of the squares of 6 consecutive natural numbers, the first number is $n^2$ and the last is $(n+5)^2$.

Prove (by induction) that for every natural $n$ that $a_n$ has remainder $7$ mod $12$.

What I did so far:

$$(n^2+(n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2+(n+5)^2)\mod 12=\\((n+1)^2+(n+2)^2+(n+3)^2+(n+4)^2+(n+5)^2+(n+6)^2)\mod 12$$

I have no idea how to continue, or if it's a good start at all...