Let $\mathcal{r}$ be the circle centered at $(0,0)$ with radius $10. $ the lines $x = 6$ and $y = 5$ divide $\mathcal{r}$ into four regions $\mathcal{r}_1$, $\mathcal{r}_2$, $\mathcal{r}_3$, and $\mathcal{r}_4$. Let $[\mathcal{r}_i]$ denote the area of region $\mathcal{r}_i$. If \[[\mathcal{r}_1] > [\mathcal{r}_2] > [\mathcal{r}_3] > [\mathcal{r}_4],\]then find $[\mathcal{r}_1] - [\mathcal{r}_2] - [\mathcal{r}_3] + [\mathcal{r}_4]$.