1. Adopting adaptiveness. One property of an \((r,q)\)-verifier that was not explicitly mentioned in Lecture 21 is whether its oracle queries are adaptive, e.g., whether the value of the \(i^{\text{th}}\) query depends on the results of the first \(i-1\) queries. Our proof that the PCP theorem is equivalent to the existence of a gap-producing reduction from 3sat to max-3sat actually relies on a non-adaptive verifier (e.g. the proof locations that it queries are a fixed function of the input \(x\) and random string \(s\)). Prove that when \(q(n)=O(1)\) this distinction is not very important: every adaptive \((r,q)\)-verifier can be simulated by a nonadaptive \((r,2^{q})\) verifier.