One subject, compound object phrases - comma or no comma?

My question is about how to punctuate sentences like the following:

The system allows searching for variables using their long name and cryptic variable names and filtering the data sets based on a given variable availability.

subject: the system

main verb: allows

object 1 (gerund+participial phrase?): searching for variables using their long name and cryptic variable names

object 2 (gerund+past-participle-phrase?): filtering the data sets based on a given variability

coordinating conjunction between object 1 and 2: the second of the two ‘and’ words

Because the second ‘and’ is not followed by a subject + verb, this sentence appears not to require a comma. However, the presence of the first ‘and’, plus the string of gerunds and nouns this sentence contains, might cause the reader to have difficulty identifying the second object phrase. I encounter many such sentences in the editing I do, but I’m loathe to rephrase too many of the ones that might cause confusion.

Hoping someone can help!