
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 an isosceles triangle  $ABC$  with side lengths  $AB = AC = 10\sqrt{2}$  and  $BC =10\sqrt{3}$ .  Construct semicircles  $P$ ,  $Q$ , and  $R$  with diameters  $AB$ ,  $AC$ ,  $BC$  respectively, such that the plane of each semicircle is perpendicular to the plane of  $ABC$ , and all semicircles are on the same side of plane  $ABC$  as shown.  There exists a plane above triangle  $ABC$  that is tangent to all three semicircles  $P$ ,  $Q$ ,  $R$  at the points  $D$ ,  $E$ , and  $F$  respectively, as shown in the diagram.  Calculate, with proof, the area of triangle  $DEF$ .
[asy]
size(200);
import three; defaultpen(linewidth(0.7)+fontsize(10)); currentprojection = orthographic(0,4,2.5); 
// 1.15 x-scale distortion factor 
triple A = (0,0,0), B = (75^.5/1.15,-125^.5,0), C = (-75^.5/1.15,-125^.5,0), D = (A+B)/2 + (0,0,abs((B-A)/2)), E = (A+C)/2 + (0,0,abs((C-A)/2)), F = (C+B)/2 + (0,0,abs((B-C)/2)); 
draw(D--E--F--cycle); draw(B--A--C); 
// approximate guess for r
real r = 1.38; draw(B--(r*B+C)/(1+r)^^(B+r*C)/(1+r)--C,linetype("4 4")); draw((B+r*C)/(1+r)--(r*B+C)/(1+r));
// lazy so I'll draw six arcs
draw(arc((A+B)/2,A,D)); draw(arc((A+B)/2,D,B)); draw(arc((A+C)/2,E,A)); draw(arc((A+C)/2,E,C)); draw(arc((C+B)/2,F,B)); draw(arc((C+B)/2,F,C)); 
label(" $A$ ",A,S); label(" $B$ ",B,W); label(" $C$ ",C,plain.E);
label(" $D$ ",D,SW); label(" $E$ ",E,SE); label(" $F$ ",F,N);[/asy]

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