
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
Three lines are drawn parallel to each of the three sides of  $\triangle ABC$  so that the three lines intersect in the interior of  $ABC$ . The resulting three smaller triangles have areas  $1$ ,  $4$ , and  $9$ . Find the area of  $\triangle ABC$ .

[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]

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