Consider a sequence $a_i$ defined such that $a_1=p,a_2=q,a_i=a_{i-1} \oplus a_{i-2}+1$ where $\oplus$ is the bitwise xor operation. How to give an upper bound of $a_n$ as a function of $p,q,n$?
In fact, there are lots of $p,q$ which satisfy $\mathop {\lim }\limits_{n \to \infty } {a_n} = \infty $, but I have found by experimetation that $a_n=O(n)$. I donnot have any idea to prove it and neither do I know whether it is correct.
If such problem is difficult to solve, I just only need to prove a specific situation: when $p,q,n \le 1000$, for each $1 \le i \le n$, $a_i \le 2^{12}$.

Consider a sequence $a_i$ defined by $$ \begin{align*} a_1&=p,\\ a_2&=q,\\ a_i&=a_{i-1} \oplus a_{i-2}+1, \end{align*}$$ where $\oplus$ is the bitwise xor operation. How can we give an upper bound for $a_n$ as a function of $p,q,n$?

There are lots of $p,q$ which satisfy $\mathop {\lim }\limits_{n \to \infty } {a_n} = \infty $, but it seems from experimentation that $a_n=O(n)$. I do not have any idea how to prove it, nor do I know whether it is correct.
If this problem is difficult to solve, I only need to prove that, for each $1 \le i \le n$, we have $a_i \le 2^{12}$ when $p,q,n \le 1000$.