An $m×n$ maze is an $m×n$ 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 9×12 maze and a 15×20 maze:
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 as a string 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 the string `1.2346e12`.