Let $X,S_t$ be $n-$dimensional random vectors such that $X\in B= \left\{ \begin{bmatrix}1\\0\\\vdots\\0 \end{bmatrix},\begin{bmatrix}0\\1\\\vdots\\0 \end{bmatrix},\cdots,\begin{bmatrix}0\\0\\\vdots\\1 \end{bmatrix} \right\}$. Let $A$ be a linear transformation matrix which when applied to a vector sums all the bits modulo $2$, appends that to the vector and finally pops the first bit. For example, if $n=4$ and $x=[1,0,0,0]^T$, then $Ax=[0,0,0,1]^T$. $S_t$ is a discrete time process where $S_0=[0,0,\cdots,0]^T$ and
$$S_{t+1}=AX+AS_t$$
where at time step $t+1$, we choose a $X$ randomly from $B$ and $S_t$ also randomly from the set of possible values of $S_t$.
I am trying to find the size of the set of possible values of $S_t$ for all $t$? If finding exact value is hard, bounds on the size will help.
Example: Let $n=4$. For simplicity, let me denote vectors by bitstrings. So $X\in \{0001,0010,0100,1000\}$, $AX\in\{0011,0101,1001,0001\}$ and $S_0=0000$. Thus,
$$S_1\in C=\{0011,0101,1001,0001\}\text{ and hence the size of set of possible values for $S_1$ is $4$}$$
Similarly, $S_2=AS_1+AX$ and since $S_1\in C$, we have $AS_1\in D=\{0110,1010,0010,0011\}$. So if we add any $x\in AS_1$ to any $y\in AX$, we get $S_2$ as follows:
$$S_2\in \{0101,0011,1111,0111,1001,1011,0001,0000,0110,1010,0010\}$$
and hence the size of set of possible values of $S_2$ is $11$. I am not seeing a pattern in which the size of possible values of $S_t$, for a general $n$, grows. Any ideas?