$1.(100$ points$)$ Considering $\backslash ((58\backslash$quad$)\_\{10\}=(\backslash$text $\{$ abcdefg $\})\_\{2\}\backslash ),$ design a synchronous sequence detector circuit that detects 'abcdefg' from a serial input stream of one bit applied to the input of the circuit with each active clock edge. The sequence detector should not detect overlapping sequences.

$($a$)(30$ points$)$ Derive the state diagram and clearly describe the meaning of each state. Specify the type of sequential circuit $($Mealy or Moore$).$

$($b$)(10$ points$)$ Determine the number of state variables to use and assign binary codes to the states in the state diagram.

$($c$)(10$ points$)$ Choose JK FFs for the implementation. Give the complete state table of the sequence detector using the reverse characteristics table of JK FF$.$

$($d$)(30$ points$)$ Obtain Boolean functions for state inputs. Also, obtain the output Boolean expression.

$($e$)(20$ points$)$ Draw the corresponding logic circuit diagram for the sequence detector.