
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
Line  $\ell$  passes through  $A$  and into the interior of the equilateral triangle  $ABC$ .   $D$  and  $E$  are the orthogonal projections of  $B$  and  $C$  onto  $\ell$  respectively.  If  $DE=1$  and  $2BD=CE$ , then the area of  $ABC$  can be expressed as  $m\sqrt n$ , where  $m$  and  $n$  are positive integers and  $n$  is not divisible by the square of any prime.  Determine  $m+n$ .

[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy]

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