Design the logic circuit that adds $3_{10}$ to a $3-$bit binary input $AB{C}_2$ providing a $3-$bit output $DE{F}_2$ that is limited to a maximum value of $7_{10}.$ Equations only $($in terms of conventional logic operators$)$ are required for the three outputs $D,E$ and $F$ described as:

DEF$=\{\begin{array}{ccc}\text{A}\text{B}\text{C}& +3_{10},\text{A}\text{B}\text{C}& <msub\; id="EditorElement\_7808651"><mrow><mn\; id="EditorElement\_7808643">4</mn></mrow><mrow\; id="EditorElement\_7808650"><mn\; id="EditorElement\_7808646">1</mn><mn\; id="EditorElement\_7808647">0</mn></mrow></msub>\\ 7_{10},& \text{A}\text{B}\text{C}& 4_{10}\end{array}$