
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
Circles  $\mathcal{P}$  and  $\mathcal{Q}$  have radii  $1$  and  $4$ , respectively, and are externally tangent at point  $A$ . Point  $B$  is on  $\mathcal{P}$  and point  $C$  is on  $\mathcal{Q}$  so that line  $BC$  is a common external tangent of the two circles. A line  $\ell$  through  $A$  intersects  $\mathcal{P}$  again at  $D$  and intersects  $\mathcal{Q}$  again at  $E$ . Points  $B$  and  $C$  lie on the same side of  $\ell$ , and the areas of  $\triangle DBA$  and  $\triangle ACE$  are equal. This common area is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

[asy]
import cse5;
pathpen=black; pointpen=black;
size(6cm);

pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);

filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
D(CR((0,1),1)); D(CR((4,4),4,150,390));
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
D(MP("E",E,N));
[/asy]

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