Zen is trying to tell us that real intuition happens when we abandon the kind of thinking that created the problem in the first place. This is especially true with koans. They trap us into relying on our typical way of thinking that creates the problem. Ultimately, the only way to solve the koan is to let go of that thinking.
Koans are either good or not so good depending on whether or not they engage us into relying on our typical way of thinking which we, ultimately, must learn to transcend. It is like we have this illness that doesn’t get any better. In fact, everything that we do only prolongs the illness and in some cases makes it even worse. The antidote that Zen proposes is to drop the way of thinking which keeps making this illness.
Part of accomplishing this requires deep immersion. This process doesn’t require much explanation except that one has to be involved in trying to answer the koan. Pretending to answer it by marathon sitting or other theatrical means will not work. Zen wants the snake to devour itself. The old ways of thinking must suddenly stop.
When the old ways of thinking suddenly stop there is no subject or object, no experiencer or experience. Consciousness (vijñāna) suddenly comes to a halt. We become the Buddha-nature suddenly and unexpectedly.