Jayarava rightly pointed out (see here) that Bodhisattvas must be able to conceive thoughts, since else no interactions would be possible between them and the 'normal' people, be it through language or through actions. Hence, the paradoxical statement of the Aṣṭasāhasrikāprajñāpāramitā saying that
[bodhisattvaḥ] tenāpi bodhicittena na manyeta
even through his bodhicitta, he should not conceive thoughts (man-).
What else could it mean? The following lines explain:
What is the reason of that? His mind (citta) is a non-mind (acitta), [since] its nature is translucent.
Hence, the key to understand what the Bodhisattva should not do through his mind lies in the understanding of what his mind actually is. He can use it withouth doing the act designated by the root man- beacuse his is a translucent "non-mind". In fact, the following paragraphs keep on discussing the nature of such a non-mind. At a certain point, one can read:
avikārā […] avikalpā acittatā
The condition of non-mind is formless and conceptionless.
The vikāras are "derived forms" of the actual reality, whereas the vikalpas are assumptions one resorts to in order to make sense of something. They are conceptual in nature and hence (mostly) not ultimately true. Hence, the statement seems to mean that a non-mind is a mind which can think without needing assumed supports (that is, conventional forms). The Bodhisattva's mind thinks directly, without needing extra explanatory devices. Can this mean that it sees without thinking, with "thinking" in the sense of "constructing thoughts"?
I am extremely grateful to the readers who richly commented on my previous post on Mahāyāna texts expressing paradoxes. Read Patrick's comment if you want to compare a different understanding of the term I translated with "translucent" (prabhāsvarā).