$\left\{ {\begin{array}{*{20}{c}}
{a + b + c = x}&{}&{}\\
{4a + b + c = 2y}&{}&{}\\
{16b + c + a = 6z}&{}&{}
\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}
{a = \frac{{2y - x}}{3}}&{}&{}\\
{b = \frac{{6z - x}}{{15}}}&{}&{}\\
{c = \frac{{21x - 10y - 6z}}{{15}}}&{}&{}
\end{array}} \right.$

$ \Rightarrow \left\{ {\begin{array}{*{20}{c}}
{a + b = \frac{{10y + 6z - 6x}}{{15}}}&{}&{}\\
{b + c = \frac{{4x - 2y}}{3}}&{}&{}\\
{c + a = \frac{{16x - 6z}}{{15}}}&{}&{}
\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}}
{\frac{{a + b}}{{a + b + c}} = \frac{{10y + 6z - 6x}}{{15x}} = \frac{{2y}}{{3x}} + \frac{{2z}}{{5x}} - \frac{2}{5}}&{}&{}\\
{\frac{{b + c}}{{4a + b + c}} = \frac{{4x - 2y}}{{6y}} = \frac{{2x}}{{3y}} - \frac{1}{3}}&{}&{}\\
{\frac{{c + a}}{{16b + c + a}} = \frac{{16x - 6z}}{{90z}} = \frac{{8x}}{{45z}} - \frac{1}{{15}}}&{}&{}
\end{array}} \right.$

Từ đó, ta viết lại $P$ thành:

$P=\left ( \dfrac{2x}{3y}+\dfrac{2y}{3x} \right )+\left ( \dfrac{2z}{5x}+\dfrac{8x}{45z} \right )-\dfrac{4}{5}\geq \dfrac{4}{3}+\dfrac{8}{15}-\dfrac{4}{5}=\dfrac{16}{15}$

Vậy GTNN của $P$ là $\dfrac{16}{15}$

$\left\{ {\begin{array}{*{20}{c}}

{a + b + c = x}&{}&{}\\

{4a + b + c = 2y}&{}&{}\\

{16b + c + a = 6z}&{}&{}

\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}

{a = \dfrac{{2y - x}}{3}}&{}&{}\\

{b = \dfrac{{6z - x}}{{15}}}&{}&{}\\

{c = \dfrac{{21x - 10y - 6z}}{{15}}}&{}&{}

\end{array}} \right.$

$ \Rightarrow \left\{ {\begin{array}{*{20}{c}}

{a + b = \dfrac{{10y + 6z - 6x}}{{15}}}&{}&{}\\

{b + c = \dfrac{{4x - 2y}}{3}}&{}&{}\\

{c + a = \dfrac{{16x - 6z}}{{15}}}&{}&{}

\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}}

{\dfrac{{a + b}}{{a + b + c}} = \dfrac{{10y + 6z - 6x}}{{15x}} = \dfrac{{2y}}{{3x}} + \dfrac{{2z}}{{5x}} - \dfrac{2}{5}}&{}&{}\\

{\dfrac{{b + c}}{{4a + b + c}} = \dfrac{{4x - 2y}}{{6y}} = \dfrac{{2x}}{{3y}} - \dfrac{1}{3}}&{}&{}\\

{\dfrac{{c + a}}{{16b + c + a}} = \dfrac{{16x - 6z}}{{90z}} = \dfrac{{8x}}{{45z}} - \dfrac{1}{{15}}}&{}&{}

\end{array}} \right.$

Từ đó, ta viết lại $P$ thành:

$P=\left ( \dfrac{2x}{3y}+\dfrac{2y}{3x} \right )+\left ( \dfrac{2z}{5x}+\dfrac{8x}{45z} \right )-\dfrac{4}{5}\geq \dfrac{4}{3}+\dfrac{8}{15}-\dfrac{4}{5}=\dfrac{16}{15}$

Vậy GTNN của $P$ là $\dfrac{16}{15}$