A pyramid has a square base $ABCD$ and vertex $E$. The area of square $ABCD$ is $1296$, and the areas of $\triangle ABE$ and $\triangle CDE$ are $450$ and $522$, respectively. What is the volume of the pyramid?

1. The base has side $36$. The altitudes of the faces $ABE$ and $CDE$ from $E$ are $\frac{2\cdot450}{36}=25$ and $\frac{2\cdot522}{36}=29$.
2. Let $E$ lie at height $z$ above a point of the base plane that is at distance $y$ from line $AB$ and $36-y$ from line $CD$. Then $y^2+z^2=25^2$ and $(36-y)^2+z^2=29^2$.
3. Subtracting: $(36-y)^2-y^2=841-625=216$, so $1296-72y=216$ and $y=15$. Then $z^2=625-225=400$, $z=20$.
4. $V=\frac13\cdot1296\cdot20=\boxed{8640}$.