How to factorize $N$ in OU cryptosystem under chosen ciphertext attack?

OU cryptosystem: $N = p^2q$, the secret key is the factorization of $N$, the public key is $g\leftarrow \mathbb{Z}_N$, $g^{p-1} \neq 1\mod p^2$, to encrypt an element $m $ $\in$ $\mathbb{Z}_p$, choose $r\leftarrow \mathbb{Z}_N$, then $Enc(m)= g^m\cdot h^r$, where $h = g^n$

In paper: Paillier's Cryptosystem Revisited [CCS01], is said that

How to factor $N=p^2q$ under CCA model?

Recall that the Okamoto-Uchiyama system consists of private key large primes $p$ and $q$, public key modulus $N=p^2q$ and elements $g,h\in \mathbb Z/N\mathbb Z$ such that $g^{p-1}\neq 1\pmod{p^2}$ and $h$ which is a non-trivial $N$th power.

Encryption for messages $0\le m<p$ is accomplished by selecting a random blinding value $r$ and computing $g^mh^r\mod N$.

If however we take a value $s>p$ we obtain an encryption of $s\mod p$. This means that we can by choose a large $s$ value (say $s>N^{1/2}$) and compute the spoof ciphertext $c=g^sh^r\mod N$. Submitting the spoof to our decryption oracle returns the value $s'=s\mod p$ so that $p|s-s'$ and by taking $\mathrm{GCD}(s-s',N)$ we recover $p$.

ETA: Perhaps more amusingly, if we compute $g^{-1}\pmod N$ via the extended Euclidean algorithm and our oracle is especially remiss in checking responses, then it will return the decryption $p-1$.