A positive integer $n$ is said to be $m$-free if $n \leq m!$ and $\gcd(i,n)=1$ for each $i=1,2,...,m$. Let $\mathcal{S}_k$ denote the sum of the squares of all the $k$-free integers. Find the remainder when $\mathcal{S}_7-\mathcal{S}_6$ is divided by $1000$.

Let's think step by step and output the final answer within \boxed{}.
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