I don't see a clear verbal vs. non-verbal dichotomy - or at least the non-verbal side has lots of variants. To gain an intuitive non-verbal understanding can involve

• visual aids (from precise to vague): graphs, diagrams, patterns (esp. repetitions), pictures, vivid imagination (esp. for memorizing)

• acoustic aids: rhythms (works with muscle memory too), patterns in the spoken form, creating sounds for elements

• abstract thinking (from precise to vague): logical inference, semantic relationships (is-a, exists, always), vague relationships (discovering that the more of this seems to imply the more of that)

Note: Logical inference seems to be the verbal part you mean, but I don't think symbolic thinking is always verbal. Its conscious derivation may be though.

And I hear that the verbal side despite lending itself to more symbolic thinking can nonetheless work its grammar magic on an intuitive level too (though not for me).

Personally if I really want to solve a mathematical problem I immerse myself in it. I try lots of attack angles from the list above (not systematically but as it seems fit). I'm an abstract thinker and don't rely on verbal, acoustic or motor cues a lot. Even visual aids don't play a large role though I do a lot of sketching, listing/enumerating combinations, drawing relations/trees, tabulating values/items. If I suspect a repeating pattern I may tap to it to sound it out. If there is lengthy logical inference involved that I haven't internalized I speak the rule repeatedly to use the acoustic loop as memory aid. I play around with it during the day visualizing relationships or following steps, sometimes until in the evening everyting blurs and I fall asleep.