An $m\times n$ maze is an $m\times n$ rectangular grid with walls placed between grid cells such that there is exactly one path from the top-left square to any other square.
The following are examples of a $9\times 12$ maze and a $15\times 20$ maze:

Let $C(m,n)$ be the number of distinct $m\times n$ mazes. Mazes which can be formed by rotation and reflection from another maze are considered distinct.

It can be verified that $C(1,1)=1$, $C(2,2)=4$, $C(3,4)=2415$, and $C(9,12)=2.5720\mathrm{e}46$ (in scientific notation rounded to $5$ significant digits).
Find $C(100,500)$ and write your answer in scientific notation rounded to $5$ significant digits.

When giving your answer, use a lowercase e to separate mantissa and exponent. E.g. if the answer is $1234567891011$ then the answer format would be 1.2346e12.