So in bitcoin proof of work mechanism, whoever solves the math equation fastest gets to mine the block. If thats the case, this miner has a lot of say in terms of the direction of newly mined transactions. Like prioritizing certain user sent transactions. Additionally since this miner is the first to mine, can't he/she be a bad actor? How can other miners make sure they aren't bad actors? Are other miners also required to process the the block after the first miner finished?
In the following text from webite:
"If any member of the network attempts to broadcast false information, all nodes on the network will immediately recognize it as objectively invalid and ignore it. Because each node can verify all information on the Bitcoin network itself, there is no need to trust other members of the network, making Bitcoin a trustless system."
I'm just confused on how other miners are able to validate and confirm a bad actor who continuous solve the math equation first. And in the above statement, when it say all nodes can verify, what does that mean? How do you define valid verses invalid?