If $a$, $b$, $c$, $d$ are positive integers, find the minimum value of $$P = \left(a + b + c + d\right)\left(\frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d}\right)$$ and the values of $a$, $b$, $c$, $d$ when it is reached.
My try: $$\left. \begin{array}{l} a + b + c + d \ge 4\sqrt[4]{abcd}\\ \frac{1}{a} + \frac{1}{b} + \frac{4}{c} + \frac{16}{d} \ge 4\sqrt[4]{\frac{64}{abcd}} \end{array} \right\} \Rightarrow P \ge 32\sqrt{2}$$
I have used mean inequalities, but that doesn't mean that I have found the minimum value. Also, I have found a similar exercise here (exercise #5), but the author shows that $P \ge 64$, which is greater than what have I found.
Can you help me solve the problem, please? Thanks!