As far as I understand, mining works like this.

They get the "data" from the pool, in in order to verify them and put them in a "block",

they try to find the nonce with which if hashed with the data and the previous hash data produces

the hash result that starts with 0000000000xxxxxx...xxx.

What I don't understand is, what is this "pool" where they send the data to all the nodes? Also, how do the nodes

verify the integrity?

What if the pool is compromised and sends the wrong data in the first place?

How is finding the right nonce alone guarantee the integrity of data and transactions ?

I would very appreciate if anyone could explain please.

Thank you.