Chalrytharendir is mostly right on all counts. One thing you can do to affect the results in any Scene is use a Emobond you have with someone to alter their results by +1/-1 once per scene, depending on whether your relationship is positive or negative. You can do this even when you're not at the scene physically.
It's true that whoever steals the Prize from the facility will technically have an easier time acquiring it. Part of the reason I had it as first come, first serve, was to reward whoever responded first. However, having the Prize paints a pretty big target on your back, especially when the Climax Phase comes around, and everyone wants it. To "acquire" a Prize from someone else, you need to beat them in battle and choose the Prize as your reward for winning the fight.
The order that you go in isn't that important in the long run. Since you still seem to be a bit unclear about what can be done in a scene, let me post a summary, with examples.
Drama Scene
You start off describing a scene your character is in. You can invite other characters to participate as well.
With your roll, you can get:
one LP recovered
an Emobond with another character participating in the scene
the Location of another character (does not have to be in the scene)
the Secret of another character (does not have to be in the scene)
Combat Scene:
You start off describing the scene, and who you're confronting.
Requires you to know the Location of whoever you're confronting.
Those with Emobonds with the participants can join in if they wish.
If you win, you can get:
the Location of one of the losing participants
the Secret of one of the losing participants
the ability to set one of the losing participants' Emobond towards you
the Prize, if the losing participant has it
I also dropped the book in the Discord. I'll just leave it at, if you like it, buy it.