A deterministic finite automaton (DFA) is a tuple M=(Q,Σ,δ) where Q is a finite set of states, Σ is an alphabet, and δ:Q×Σ→Q is a transition function. A synchronizing word (or reset word) for a DFA is a word w∈Σ∗ such that there exists a single state p∈Q with the property that reading w from any state q∈Q leads to p; formally, δ∗(q,w)=p for all q∈Q, where δ∗ denotes the extension of δ to words. A DFA is called synchronizing if it admits at least one synchronizing word. Prove or disprove: For every synchronizing DFA with n states, there exists a synchronizing word of length at most (n−1)2.