In triangle $ABC,$ $E$ lies on $\overline{AC}$ such that $AE:EC = 2:1,$ and $F$ lies on $\overline{AB}$ such that $AF:FB = 1:4.$  Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}.$

[asy]
unitsize(0.8 cm);

pair A, B, C, D, E, F, P;

A = (1,4);
B = (0,0);
C = (6,0);
E = interp(A,C,2/3);
F = interp(A,B,1/5);
P = extension(B,E,C,F);

draw(A--B--C--cycle);
draw(B--E);
draw(C--F);

label("$A$", A, N);
label("$B$", B, SW);
label("$C$", C, SE);
label("$E$", E, NE);
label("$F$", F, W);
label("$P$", P, S);
[/asy]

Then
\[\overrightarrow{P} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\]where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$  Enter the ordered triple $(x,y,z).$
From the given information,
\[\overrightarrow{E} = \frac{1}{3} \overrightarrow{A} + \frac{2}{3} \overrightarrow{C}\]and
\[\overrightarrow{F} = \frac{4}{5} \overrightarrow{A} + \frac{1}{5} \overrightarrow{B}.\]Isolating $\overrightarrow{A}$ in each equation, we obtain
\[\overrightarrow{A} = 3 \overrightarrow{E} - 2 \overrightarrow{C} = \frac{5 \overrightarrow{F} - \overrightarrow{B}}{4}.\]Then $12 \overrightarrow{E} - 8 \overrightarrow{C} = 5 \overrightarrow{F} - \overrightarrow{B},$ so $12 \overrightarrow{E} + \overrightarrow{B} = 5 \overrightarrow{F} + 8 \overrightarrow{C},$ or
\[\frac{12}{13} \overrightarrow{E} + \frac{1}{13} \overrightarrow{B} = \frac{5}{13} \overrightarrow{F} + \frac{8}{13} \overrightarrow{C}.\]Since the coefficients on both sides of the equation add up to 1, the vector on the left side lies on line $BE,$ and the vector on the right side lies on line $CF.$  Therefore, this common vector is $\overrightarrow{P}.$  Then
\begin{align*}
\overrightarrow{P} &= \frac{12}{13} \overrightarrow{E} + \frac{1}{13} \overrightarrow{B} \\
&= \frac{12}{13} \left( \frac{1}{3} \overrightarrow{A} + \frac{2}{3} \overrightarrow{C} \right) + \frac{1}{13} \overrightarrow{B} \\
&= \frac{4}{13} \overrightarrow{A} + \frac{1}{13} \overrightarrow{B} + \frac{8}{13} \overrightarrow{C}.
\end{align*}Thus, $(x,y,z) = \boxed{\left( \frac{4}{13}, \frac{1}{13}, \frac{8}{13} \right)}.$