Eccoci.

In this problem we consider a random walk on the integers $\mathbb{Z}$, in which our position at time $t$ is denoted as $X_t$.

At time $0$ we start at position $0$. That is, $X_0=0$.
At time $1$ we jump to position $1$. That is, $X_1=1$.
Thereafter, at time $t=2,3,\dots$ we make a jump of size $|X_{t-2}|$ in either the positive or negative direction, with probability $1/2$ each way. If $X_{t-2}=0$ we stay put at time $t$.

At $t=5$ we find our position $X_5$ has the following distribution: $$ X_5=\begin{cases} -1\quad&\text{with probability }3/8\\ 1\quad&\text{with probability }3/8\\ 3\quad&\text{with probability }1/8\\ 5\quad&\text{with probability }1/8\\ \end{cases} $$ The standard deviation $\sigma$ of a random variable $X$ with mean $\mu$ is defined as $$ \sigma=\sqrt{\mathbb{E}[X^2]-\mu^2} $$ Furthermore the skewness of $X$ is defined as $$ \text{Skew}(X)=\mathbb{E}\biggl[\Bigl(\frac{X-\mu}{\sigma}\Bigr)^3\biggr] $$ For $X_5$, which has mean $1$ and standard deviation $2$, we find $\text{Skew}(X_5)=0.75$. You are also given $\text{Skew}(X_{10})\approx2.50997097$.

Find $\text{Skew}(X_{50})$. Give your answer rounded to eight digits after the decimal point.