How is $\text{PCP}[O(\log n),O(1)]$ NOT P?

Question

As a prover, we just try to convince the verifier that it's correct, no matter whether it is or not. So we can just analyze every possible route.

For $\text{PCP}[O(\log n),O(1)]$, won't there just be polynomial many possible routes, and checking all just cost polynomial time?

In computational complexity theory, an interactive proof system is an abstract machine that models computation as the exchange of messages between two parties. The parties, the verifier and the prover, interact by exchanging messages in order to ascertain whether a given string belongs to a language or not. The prover is all-powerful and possesses unlimited computational resources, but cannot be trusted, while the verifier has bounded computation power. Messages are sent between the verifier and prover until the verifier has an answer to the problem and has "convinced" itself that it is correct.