Converting a minimized boolean algebric expression into one that uses exclusively NANDs and NOTs.

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After creating a truth table of my function, transforming it into a Karnaugh Map and extracting the minimized expression from it, I end up with the following expression:

$$AD+\overline{A}\overline{B}\overline{C}\overline{D}+BC+BD+CD$$

The goal is to find an expression equivelant to the one above so that I can create a circuit utilizing only NAND and NOT gates.

I believe that the way I did this isn't correct so here's my process:

$$AD+BD+\overline{A}\overline{B}\overline{C}\overline{D}+BC+CD$$ Applying a double negation: $$\overline{\overline{AD+BD}}+\overline{A}\overline{B}\overline{C}\overline{D}+\overline{\overline{BC+CD}}$$ Applying the De Morgan rule: $$=\overline{\overline{AD}.\overline{BD}}+\overline{A}\overline{B}\overline{C}\overline{D}+\overline{\overline{BC}.\overline{CD}}$$ Simplifying: $$=\overline{\overline{ABD}}+\overline{\overline{BCD}}+\overline{A}\overline{B}\overline{C}\overline{D} = ABD + BCD + \overline{A}\overline{B}\overline{C}\overline{D}$$ Repeating the steps: $$=\overline{\overline{ABD+BCD}} + \overline{A}\overline{B}\overline{C}\overline{D} = \overline{\overline{ABD}.\overline{BCD}}+\overline{A}\overline{B}\overline{C}\overline{D}$$ How to continue from here so that in the end there are only ANDs and NOTs?