We shall define a sqube to be a number of the form, $p^2{q}^3$, where $p$ and $q$ are distinct primes.
For example, $200=5^2{2}^3$ or $120072949={23}^2{61}^3$.

The first five squbes are $72,108,200,392$, and $500$.

Interestingly, $200$ is also the first number for which you cannot change any single digit to make a prime; we shall call such numbers, prime-proof. The next prime-proof sqube which contains the contiguous sub-string "$200$" is $1992008$.

Find the $200$th prime-proof sqube containing the contiguous sub-string "$200$".