Novice question regarding Rivest Shamir Wagner Time Lock Puzzles of the form $x^{2^t} \bmod N $ with $ N=p.q $ primes.

Question

I'm using the Rivest Shamir Wagner Time Lock Puzzle setup in an application. The puzzles are of the form: $x^{2^t} \bmod N $ with $ N=p.q $ and p and q are primes. My question is this: assuming I pick a random x that is a square root modulo N, checking that x satisfies $\gcd(x+1,N)=1 $ as explained in this paper (https://eprint.iacr.org/2019/619) then, what is the size of the group defined by the repeated squaring of x modulo N ?