I think the question is hard to answer without teasing out a lot of presuppositions. For example:
I'm not sure Orthodoxy would agree with the statement that Mary was "born with sin", but it depends on what you mean by it. I assume it's a reference to the idea of "original sin" (or the Orthodox might say "ancestral sin") and from the Orthodox perspective it would only mean that Mary shared in the same "fallen" human nature with its capacity and propensity towards sin. That doesn't necessarily entail that she actually committed any particular sins, or was separated from God by sin. You could delve arbitrarily deeply into trying to elaborate systematic notions about sin, but I think the crux of the Orthodox position is not so much based around some sort of evaluation of the question of whether Mary ever "committed a sin", but around the purity of her openness to God and the beauty of her response to the angel in Luke 1. She had found favor with God and rejoiced in God her savior.
From a different perspective, I don't think Orthodox theology has ever attempted to explain "how" questions of the sort you are asking: "how was she purified of all sin?" Or at least not if you are looking for some kind of technical explanation. Insofar as the question also proceeds from assumptions about being "born with sin", that might not reflect an Orthodox understanding, the question might also be moot?
In any case, I would also suggest that it would be akin to asking "how" God purifies a person in order to be a temple of the Holy Spirit. In both cases, in Orthodox teaching the person is not entirely and perfectly passive. Neither Mary nor we are merely "incubators", so to speak. Mary is called blessed by all generations of Christians because she is understood to have said "yes" to God in a beautiful and unique way, which is certainly a matter of grace, but also of her response to that grace. I do not think there is a simple and systematic answer to the question of how this works. It is a transformative process in one's own life. The Spirit blows where it wills and you cannot tell where it came from or where it is going, so to speak. It is somewhat individual. There may be patterns -- the Church fathers spend a lot of time talking about working to change patterns of thinking and behavior, and how to accomplish that -- but the patterns don't amount to a technical explanation.
Orthodoxy rejoices in the beauty and purity of Mary's devotion, and in the grace she received, but doesn't attempt to systematically explain it. It might refrain from saying she was born without any possibility of sinning (as in the dogma of the Immaculate Conception) simply because to say that would seem to conflict with her own rejoicing in God as her savior. Why would she have needed a savior in such a case? But the rejection isn't so much about favoring an equally systematic alternate view, as much as wanting to avoid extrapolating too far beyond what we are justified in saying.