Given the following boolean function:

F$($A$,$B$,$C$,$D$)=$ A$\text{'}$ B$\text{'}$ C$\text{'}$ D $+$ CD $+$ A C$\text{'}$ D

$(1)$ Simplify F using K$-$map method.

$(2)$ Draw the logic diagram $($with gates$)$ of the simplified F function from part $(1).$

$(3)$ Implement the logic diagram from part$(2)$ with NAND gates.

$(4)$ Simplify the function F algebraically $($you should get the same as the simplified function from part$(1),$ please use proper algebraic rules to simplify$).$