Design a sequiantial counter that counts through $1,2,2,3,1,2,2,3..$ using a minimum number of $D$ type flip flops and only a mimimam mamber of X and $Y$ gates given below.

For the X and Y gates, Z represent high impedance. Complement of the variables or other logic gates are not available: Logic levels $1$ and $0$ are available.

a$)$ State diagram $(5$ Pointis$).$

b$)$ State table and equations. $(5$ Points$)$

c$)$ Full design $(25$ points$).$