Consider a $3-$bit up counter that starts with $000$ and counts to $111.$ After $111,$ it goes back to $000.$ Answer the following questions with using $T$ flip$-$flops. $(40$ points$)$

a$.$ Draw the state transition diagram

b$.$ Write out the state transition table and use $K-$maps to derive the equations of the next states

c$.$ Draw the circuit using T flip$-$flops

d$.$ Design this $3-$bit counter with JK flip$-$flops. Demonstrate a working prototype of it$.$

Explain how you know the circuit works. $(10$ extra credit points$)$

$3.$ points$)$

a$.$ Complete the truth table where $J,K,Q$ are the inputs and $Q_{\text{n}\text{e}\text{x}\text{t}\text{s}\text{t}\text{a}\text{t}\text{e}}$ and $T$ are the outputs.

$\backslash$table$[[J,K,,Q,Q_{\text{n}\text{e}\text{x}\text{t}\text{s}\text{t}\text{a}\text{t}\text{e}}$