$[25$p$]5.$ Derive the Boolean equations for two $4-$input logic functions, $x$ and $Y$ based on the following:

$($i$)$ The output $x$ is one when the input does not contain $3$ consecutive $1\text{'}$s in the word $A_3,A_2,A_1,$

$A_0.$ The output $x$ is zero, otherwise. Fill in the truth table and use the product of sums form to write

the corresponding Boolean equation for $x.($No simplification needed.$)$

$($ii$)$ The output $Y$ is one when no two adjacent bits in the word $A_3,A_2,A_1,A_0$ are the same $($e$.$g$.,$ if

$A_2$ is $0$ then $A_3$ and $A_1$ cannot be $0).$ The output $Y$ is zero, otherwise $($e$.$g$.,0000).$ Fill in the truth

table above and use the sum of products form to write the corresponding Boolean equation for $Y.$

$($No simplification needed.$)$

$($iii$)$ Please draw the circuit of Y using only $2-$input XOR and AND gates.