Let say we have $N=pq$ (Both $p$ and $q$ are safe primes, meaning $\gcd(p-1,q-1)=2$)
let's assume $e$ is an odd number which can be efficiently factored and assume that we know the $e$'th root of 1 modulo $N$ - call it $r$ - as well es $e$ and its prime-factorization. There is a claim that knowing the $e$'th root of 1 modulo $N$ we can find such $y>1$ and an odd prime $p_0$ such that $y^{p_0} = 1 \bmod N$.
How can we calculate $y$ and $p_0$?