Why is this cryptosystem insecure? Can someone help me see the flaw in this cryptosystem?

Note: This is homework and it is due today at 1:30pm. An answer before that is not expected; I'd just like to understand what the flaw is. 

From An Introduction to Mathematical Cryptography By Jeffrey Hoffstein, Jill Pipher and Joseph H. Silverman (Springer, 2008); problem 3.10, found one click away from here:


  Alice chooses two large primes $p$ and $q$ and she publishes $N = pq$. It is assumed
  that $N$ is hard to factor. Alice also chooses three random numbers $g$, $r_1$, and $r_2$ modulo $N$ and computes
  $$g_1\equiv g^{r_1\cdot(p−1)}\pmod N\text{ }\text{ and }\text{ }g_2\equiv g^{r_2\cdot(q−1)}\pmod N.$$
  
  Her public key is the triple $(N,g_1,g_2)$ and her private key is the pair of primes $(p,q)$.
  
  Now Bob wants to send the message $m$ to Alice, where $m$ is a number modulo $N$. He chooses two random integers $s_1$ and $s_2$ modulo $N$ and computes
  $$c_1\equiv m\cdot{g_1}^{s_1}\pmod N\text{ }\text{ and }\text{ }c_2\equiv m\cdot{g_2}^{s_2}\pmod N.$$
  Bob sends the ciphertext $(c_1,c_2)$ to Alice.
  
  Decryption is extremely fast and easy. Alice use the Chinese remainder theorem
  to solve the pair of congruences
  $$x\equiv c_1\pmod p\text{ }\text{ and }\text{ }x\equiv c_2\pmod q.$$
  
  (a) Prove that Alice’s solution $x$ is equal to Bob’s plaintext $m$.
  (b) Explain why this cryptosystem is not secure.


I can verify A but am having trouble seeing where the cryptosystem is weak.
Avenues I've explored:


The modulus is factorable
The encrypted text is easily decryptable


But I'm just not seeing the answer. Can I have some help?
From the key creators point of view, notice that:$g_1\bmod p=1$ and $g_2\bmod q=1$.

Which means that $(g_1-1)\cdot(g_2-1)\bmod N=0$.

Which implies that $g_1-1$ and $g_2-1$ share a common divisor with $p$ and $q$.

To obtain $p$ we simply take $\gcd(g_1-1,N)$.

To obtain $q$ we simply take $\gcd(g_2-1,N)$, or $N/p$.

Because we have been able to factor the modulus using only the public keys and with no protocol error on the part of the implementer, this cryptosystem is insecure.