I have a question about a sequence of modular exponentations. Suppose N is a 700-digit semiprime. Consider a sequence $a$ where $a_1$ is a random number between 1 and N, and $a_{n+1} = (a_n^d mod N)$. Given that I know some sub-sequence $a_1, a_2, a_3, ... a_p$, is it possible to predict any subsequent element in the sequence after $a_p$ without knowing the exponent $d$?
For example, let N be 10337562743 (which is not 700-digits, but this is just an example). Let's say we know $a_1, a_2, a_3$, and want to predict $a_4$.
Let $a_1$ = a random number between 1 and N. Then we have:
$$a_2 = (a_1^d mod N)$$ $$a_3 = (a_2^d mod N) ≡ a_1^{d^2} \pmod N$$
So we know a system of congruences about $a_4$:
$$a_4 = (a_3^d mod N) ≡ a_2^{d^2} ≡ a_1^{d^2}\pmod N$$
Or in other words:
$$a_4 ≡ a_3^d ≡ a_2^{d^2} ≡ a_1^{d^2} \pmod N$$
I can't think of any way to use this system of congruences to my advantage to predict the value of $a_4$ or any subsequent value. I understand why logarithms do not work with modular congruences and why solving for $d$ is difficult, but I am wondering if there are any relationships in the above problem that I am missing.
Thanks!