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We start out the episode following River around Serenity. We get a feel for her state of mind, hearing with her people's thoughts as she gets near them. Actually, I feel like she wasn't reading exact thoughts so much as a combination of thoughts and feelings, because I don't really think some of the "thoughts" made literal sense. She eventually wanders to the cargo deck, which she pictures as being full of fallen leaves. She picks up what she thinks is a stick - and then we switch to the point of view of the rest of the crew, to find out that she's holding a loaded gun. She surrenders the gun to their urgent requests, so no harm comes to anyone.
Even so, Mal holds a crew meeting about her and the safety of the ship. Simon is convinced that she would never harm anyone, and claims she never even shot a gun before. Kaylee feels compelled to tell them about how River shot the three men during the rescue of Mal in "War Stories"; Simon is stunned. Mal doesn't come to a decision on what to do about River, but he makes it clear that it's possible that he'll ask Simon and River to leave Serenity. Everyone else is worried for their safety, but they are also concerned about turning River and Simon out. River overhears the whole discussion.
That "night" when the crew heads to bed, a small, unknown ship approaches Serenity. We see the man inside looking over the arrest warrants for Simon and River. The ship comes alongside, and the man enters Serenity. He has some knowledge of Firefly ships, because he activates certain controls and locks all the rooms, which locks in the sleeping Mal, Jayne, Zoe, and Wash. He finds Kaylee working in the engine room and ties her up, and then he runs into Book and knocks him out. Finally he finds Simon, who had just discovered that River is not in her room. The man, whose name is Jubal Early, is a bounty hunter after Simon and River. He threatens Simon with his gun and makes Simon search the ship with him for River.
Early is a very odd man, as he poses questions and comments that sound deep, but really don't go anywhere. He is, however, extremely polite. He and Simon search the ship, even waking Inara and searching her shuttle, but do not find River. They end up on the bridge with Early getting extremely frustrated.
Early begins broadcasting over the ship's intercom, calling for River and telling her he's going to shoot Simon. This wakes up Mal, Zoe, and Wash (Jayne keeps sleeping), but they are still locked in. River responds by intercom, claiming that she has become the ship. Her verbal fencing with Early makes him even more frustrated and desperate. During this, we also see that River is privately communicating with Mal and Kaylee to make plans with them. She convinces Kaylee to free herself and then activate some ship systems for River. Finally, River gives away the fact that she's on Early's ship; she says she knows that no one on Serenity wants her around, so she'll willingly go with Early.
Early nearly falls over himself heading to his ship, both to keep River from hurting it and to take her into custody. Simon attacks Early, but gets shot in the leg for his trouble. River has somehow let Mal out of his room, and he attacks Early, too. Early prevails, and continues for his ship. When he gets outside of Serenity, Mal shows up in a spacesuit and shoves Early off into space. River returns to Serenity from Early's ship. Simon directs Zoe in removing the bullet from his leg, and all seems to be well.
The highlight of this episode is the dialog, with Early's pseudo-philosophical monologues and River's "impersonation" of Serenity. Frankly, River and the rest of the crew are lucky that Early was a semi-decent man who didn't decide to shoot everyone at the first sign of resistance. I find it hard to believe that the Alliance would care if the others were killed as long as River and Simon were also killed or taken into custody.
The episode seems to leave River in Mal's good graces, since she has demonstrated her concern for Mal's crew and Serenity. However, we'll never know Mal's decision concerning Simon and River remaining on Serenity, since the later feature film takes up the story sometime later. Our look at what she experiences telepathically and empathically at the beginning of the episode shows us just how mixed up she is, letting us sympathize with her more. In particular, she "reads" how Simon feels he's lost a lot by saving her; I don't think Simon would think of it that way, but obviously at least a little bit of that feeling is there in him.
A big unresolved issue is Inara leaving Serenity; she hadn't told the rest of the crew yet, but Mal is pressuring her to. However, with the events of the episode, she never does. This is a loss for the viewers, since that certainly would be a pivotal event in the series.