Adalogical Ænigma #2

Hints on Solving the Ænigma

Gracious greetings, gentle patron!
If you are reading this page, then I fear you may have ceased making progress on my second ænigma and are perhaps on the verge of declaring defeat. I must encourage you, in the strongest possible terms, not to yield to despair, to rather take heart and allow me to give you a small hint or two. It may be, with just a modicum of my assistance, that you will soon be taking my ænigma by storm!
Regrettably, I cannot with certainty predict as I pen these words exactly how far into the solution you have yet traveled. If you have not yet made any progress whatsoever, then I suggest you try reading through my detailed walkthrough of the example puzzle first; it should provide you with some guidance and commonly used solving techniques.
Otherwise, I think it likely that you will have divined that the most efficacious starting point for solution is in the lower-right corner, in the region labelled with the numeral ‘6’. Indeed, I hope that the techniques shewn in the example walkthrough will have allowed you to progress at least as far as the situation depicted below:
(I have introduced labels for two of the regions of the grid, for ease of reference in what follows.)
We must concentrate our attentions on the vicinity of region ‘A’. The left end of the part of the path passing through region ‘B’ has already entered region ‘A’. Now consider the part of the path passing through the region labelled ‘6’; although we are currently unsure of how that end will continue, it is clear that it must next also enter region ‘A’.
Recall that the loop must enter and leave each region exactly once. We can therefore conclude that the two path ends we've just discussed must join with one another very shortly indeed, somewhere inside region ‘A’. Even though we cannot yet see exactly how they will meet, we can nonetheless take that meeting for granted and make further inferences on that basis.
Now, I pray that you take a pause to see if this reasoning does not allow you to make substantial further progress on your own. (Indeed, this style of inference will come in handy on a number of occasions in the remainder of the ænigma, so you would be well advised to keep a wary eye out for its possible use.)
If you have taken this advice to heart, but have nonetheless found yourself stuck once again at some later stage of the solution, I suggest that you proceed to the next page of hints.