
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
[asy]
unitsize(27);
defaultpen(linewidth(.8pt)+fontsize(10pt));
pair A,B,C,D,E,F,X,Y,Z;
A=(3,3); B=(0,0); C=(6,0);
D=(3,0); // Placing D midway between B and C
E=(3,2); // Placing E such that BE cuts AC into half
F=(1.5,1.5); // Placement of F dividing AB into two equal segments
draw(A--B--C--cycle);
draw(A--D);
draw(B--E);
draw(C--F);
X=intersectionpoint(A--D,C--F); // Point N1
Y=intersectionpoint(B--E,A--D); // Point N2
Z=intersectionpoint(B--E,C--F); // Point N3
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,S);
label("$E$",E,NE);
label("$F$",F,NW);
label("$N_1$",X,NE);
label("$N_2$",Y,NE);
label("$N_3$",Z,S);
[/asy]
The lines $\overline{CD}$, $\overline{AE}$, and $\overline{BF}$ divide their respective sides into equal halves. With intersections as shown, find the ratio of the area of triangle $N_1N_2N_3$ to triangle $ABC$.

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