There are $k$ different types of characters in Tuta-Puta language. You start adding these characters to an empty string. At every addition, you choose any one of these $k$ characters with equal probability. You stop when every type of character occurs odd number of times in the string.

What is the expected length of your string if $k=2282023$ ?

Given that the answer is of the form $\frac{p}{q}$, where the greatest common divisor of $p$ and $q$ is $1$, find the value of $x$ where $x \equiv p \cdot q^{-1} \text{mod } 10^9+7$ and $0 \leq x < 10^9+7$.