Rand48 is a pseudorandom number generator used by some programming languages. It generates a sequence from any given integer $0\le a_0<2^{48}$ using the rule $a_n=(25214903917\cdot a_{n-1}+11)mod{2}^{48}$.

Let $b_n=\lfloor a_n/2^{16}\rfloor mod52$. The sequence $b_0,b_1,\dots$ is translated to an infinite string $c=c_0{c}_1\dots$ via the rule:
$0\to$ a, $1\to$ b, $\dots$, $25\to$ z, $26\to$ A, $27\to$ B, $\dots$, $51\to$ Z.

For example, if we choose $a_0=123456$, then the string $c$ starts with: "bQYicNGCY$\dots$".
Moreover, starting from index $100$, we encounter the substring "RxqLBfWzv" for the first time.

Alternatively, if $c$ starts with "EULERcats$\dots$", then $a_0$ must be $78580612777175$.

Now suppose that the string $c$ starts with "PuzzleOne$\dots$".
Find the starting index of the first occurrence of the substring "LuckyText" in $c$.