
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
Consider triangle $DEF$ where $F$ is on the plane, $DF=39$, $EF=42$, and $P$ is a point on $DE$ such that line $FP$ bisects angle $DFE$. Find the ratio of the area of triangle $DFP$ to the area of triangle $EFP$. Use the following diagram as a guide:
[asy]
import markers;

real t=42/(39+42);
pair D=(-20,0);
pair E=(21,0);
pair F=(0,30);
pair P=t*D+(1-t)*E;

draw(F--D--E--F--P);

label("$D$",D,SW);
label("$E$",E,E);
label("$F$",F,N);
label("$P$",P,NE);

label("$39$",.5*(D+F),NW);
label("$42$",.5*(E+F),NE);
[/asy]

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