Design a sequence recognizer, using a minimum number of D$-$flipflops, that can detect the sequence $"1001".$ The recognizer circuit has only one input $(x)$ where one bit of input is supplied on every clock cycle. There is one output $($Y$)$ that becomes $"1"$ when the desired pattern is detected. Make sure to include all design steps such as the state diagram, state table, K$-$maps, and Boolean expressions.

The following example that shows the detection of the desired bit pattern $"1001"$:

Input $(x)$:$,11100110100100110$dots

Output $($Y$)$: $00000100000100100$dots $($Y becomes $1$ when the pattern is detected$)$

Note:

If the solution produces correct results but doesn't use a minimum number of D Flip$-$Flops, you will lose $\frac{1}{3}$ of the total weight $($i$.$e$.,5$ points deduction$).$

If the solution doesn't produce correct results, the deduction will be based on multiple factors including the procedure, number of correct steps, etc.