Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.
Notice that the quadratic polynomial $p(x) = 1 + x + \frac{1}{2}x(x-1)$ satisfies $p(j) = 2^{j}$ for $j = 0, 1$ and $2$. A polynomial $q(x)$ of degree 7 satisfies $q(j) = 2^{j}$ for $j = 0, 1, 2, 3, 4, 5, 6, 7$. Find the value of $q(10)$.