What will happen if we add one more gate to the flip flop?
What you've drawn there is a flip-flop-flup. But as it turns out, it's not all that useful or usable. It can be extended further, into an N-flop, and perhaps it's as easy to analyse in the N form as the 3 form.

Consider the following. 

Let all inputs be low. All outputs will therefore be high. Any gates sees high and low inputs, so the output is high. This is stable.

Let all inputs be high. If any output is low, then all gates fed by that output will be forced high. All inputs at the low gate are high, so it will stay low. One low state is stable with all inputs high. Mercy sakes alive, look's like we got us an N-flop! (apologies to Convoy).

With all other inputs high, let one input go low. That will force its output high, but it was high anyway, so it hasn't affected the stable low output on a different gate. If we want to set a different stable state, then we have to take all inputs low, and leave a high on the one we want to set. This will flip the state. To retain that state, we now take all the other inputs high at the same time. This needs either more gating, or a multi-way radio switch. So we haven't got a complete solution.

So we can figure out what this circuit does, but it's just not very useful. I guess if it was useful, it would be in the literature as a neat concept, and it isn't.

As an exercise, how would we make a useful N-flop, or radio button circuit? It would require only one input to change state between the holding and the setting conditions, rather than N-1 of them. The logic equation for each 'state' would therefore be, be true if 'set input' OR NOT (any other states true). You can immediately see that this has two levels of gating. If you like, it's introduced the second level of gating that would be needed to make the above attempt at an N-flop work nicely.