Design a mod$-4$ counter that counts down in numerical order when a $1-$bit input called mode is low. When mode is high, the counter counts up in Gray code order. Use JK flip$-$flops. The machine also has an output called detect that is high whenever the present value of the counter is an odd number. Match each of the next state variables J$0,$ KO$,$ J$1,$ K$1,$ and the output detect with its correct Boolean expression.

Notation $-$ apostrophe indicates inversion: $A^{\text{'}}=$not$(A)$

J$0$

KO$.$

J$1$

K$1$

detect