
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In triangle $ABC$, $D$ lies on $\overline{BC}$ extended past $C$ such that $BD:DC = 2:1,$ and $E$ lies on $\overline{AC}$ such that $AE:EC = 2:1.$ Let $P$ be the intersection of lines $BE$ and $AD.$

[asy]
unitsize(0.8 cm);

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

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

draw(A--B--C--cycle);
draw(A--D--C);
draw(B--P);

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

Find the parametric equation of point $P$ in terms of $\overrightarrow{A},$ $\overrightarrow{B},$ and $\overrightarrow{C}$ such that $\overrightarrow{P} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},$ where $x + y + z = 1.$ Determine the values of $x, y,$ and $z$.

Present the answer in LaTex format: \boxed{Your answer}
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