When did Satan enter into Judas?

Judas is remembered primarily for his betrayal of Jesus. Two gospels suggest that Judas betrayed Jesus under the influence of Satan, that Satan had “entered into him”. They disagree, however, about when this influence took place.

All three of the synoptic gospels describe Judas going to the chief priests offering to betray Jesus to them. Luke adds a detail not included in the other two gospels that helps to explain his behaviour, saying that Satan had entered into Judas:

Now the festival of Unleavened Bread, which is called the Passover, was near. The chief priests and the scribes were looking for a way to put Jesus to death, for they were afraid of the people. Then Satan entered into Judas called Iscariot, who was one of the twelve; he went away and conferred with the chief priests and officers of the temple police about how he might betray him to them. They were greatly pleased and agreed to give him money. So he consented and began to look for an opportunity to betray him to them when no crowd was present. [Luke 22:1-6 (NRSV)]

In Luke, Satan thus enters into Judas and prompts him to arrange to betray Jesus.

In John’s account of the betrayal, though, Satan enters into Judas later, at the Last Supper, immediately before he actually betrays Jesus:

After saying this Jesus was troubled in spirit, and declared, ‘Very truly, I tell you, one of you will betray me.’ The disciples looked at one another, uncertain of whom he was speaking. One of his disciples — the one whom Jesus loved — was reclining next to him; Simon Peter therefore motioned to him to ask Jesus of whom he was speaking. So while reclining next to Jesus, he asked him, ‘Lord, who is it?’ Jesus answered, ‘It is the one to whom I give this piece of bread when I have dipped it in the dish.’ So when he had dipped the piece of bread, he gave it to Judas son of Simon Iscariot. After he received the piece of bread, Satan entered into him. Jesus said to him, ‘Do quickly what you are going to do.’ [John 13:21-27 (NRSV)]

So did Satan enter into Judas before he arranged to betray Jesus, or before he actually betrayed Jesus?