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I am trying to understand the concept of synchronised DFAs. I have a question where all the states in the DFA after reading that particular word from the alphabet will reach a particular state with the alphabet (A to Z). And I should prove that a language which accepts any random number of alphabet combinations followed by that particular word is either fully accepted by that synchronised DFA or it is not accepted at all.

I know my question might be confusing but maybe someone can explain to me like a idea how should I start because I'm not really sure and confused about how to go about here.

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A synchronizing word $w$ has the property that there is a single state $q$ of the DFA, such that for every state $s$ it holds that $\delta(s,w)=q$. That is, after reading $w$ you end up in $q$, regardless of where you started.

Now, suppose you read a bunch of letters. You know nothing about them. But you know that after reading them you are going to read a synchornizing word. Where will the automaton be after reading the entire sequence?

Can you take it from there?

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