You don't study scaring, you just do it.
I'm not even breaking a sweat.
Archon’s Palace wrote:
Interrupt: When your opponent wins a command struggle, exhaust this support to cancel either the card bonus or the resource bonus of that planet this phase.
But the important part is about the rules regarding Void Pirates, Rogue Traders, and Eldar Survivalist
From page 12 of the rules ref guide.
The +1(cards) and +1(resources) symbols modify the number of cards or resources that are granted by a planet when the controller of the symbol wins a command struggle there.
So they modify the planet and Archon's Palace cancels either the card bonus or the resource bonus of that planet.