If $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \frac{p(x)}{x(1-x)}$ . If $q(x) = q\left(\frac{1}{1-x}\right)$ for every $x \ne 0$, and $p(2) = -7, p(3) = -11$, find $p(10)$.

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