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Sketch a deterministic finite automaton (DFA) for the following regular language (type 3):$$ L ~=~ \{ w \in {0,1}^* ~:~ |w|_0 \equiv |w|_1 \text{ mod } 2 \} $$
This is an infinite language where \(|w|_0\) (number of 0s in the word \(w\)) and \(|w|_1\) (number of 1s in the word \(w\)) are both either even or odd:$$ L ~=~ \{ \varepsilon, 00, 11, 01, 10, 1111, ~... \} $$