Analysis does not need to be informative if there is always another kind of knowledge for it to act upon.
For 'a posteriori' knowledge, we can obviously form referents by actually indicating real objects, so no analysis needs to be informative. Analysis can then be applied to information, and does not have to provide information itself.
But it is clear that some knowledge is 'a prior', that we have leverage on information from some basis that is integral to our minds. Even to grasp what is pointed to when someone points at a table and calls it a table, one needs a set of underlying organizing conventions that limit the range of the potential implicit pronoun created by pointing. And even below that, one needs to know that pointing is. This set of conventions allows analysis to get a foothold by the process of elimination. But if these underlying conventions do not limit the options to a finite set, no information really can be conveyed.
Many people accept Kant's notion that synthetic a priori knowledge is indeed possible. There seems to be formational content present for all of us that provides enough structure for us to learn meanings from tacit references and implicit indications.
Plato's notion of anamnesis suggests that, at base, all knowledge is simply recombinations of this kind of knowlege -- that we already know everything we can learn, but need to apply analysis. So in extreme models of the mind, one does not ever have to be able to generate new synthetic knowledge. One only has to trust that it is there.
Most thinkers will not go that far, but they will go part of the way. From a naturalist point of view, we are evolved beings and we are formed from information, in the form of genetic code. So it is unlikely that none of that information ends up in our minds at birth as a basis for bootstrapping learning.
So the paradox is not really paradoxical. Analysis may be thoroughly incapable of generating information on its own. But it does not need to. Synthetic knowledge can be the ultimate source of all information, with analysis providing a different aspect of understanding than 'information'.