Vector Algebra

Let \(\vec{a}=9 \hat{i}-13 \hat{j}+25 \hat{k}, \vec{b}=3 \hat{i}+7 \hat{j}-13 \hat{k}\) and \(\vec{c}=17 \hat{i}-2 \hat{j}+\hat{k}\) be three given vectors. If \(\vec{r}\) is a vector such that \(\vec{r} \times \vec{a}=(\vec{b}+\vec{c}) \times \vec{a}\) and \(\vec{r} \cdot(\vec{b}-\vec{c})=0\), then \(\frac{|593 \vec{r}+67 \vec{a}|^2}{(593)^2}\) is equal to __________.