
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
Two congruent equilateral triangles  $\triangle ABC$  and  $\triangle DEF$  lie on the same side of line  $BC$  so that  $B$ ,  $C$ ,  $E$ , and  $F$  are collinear as shown. A line intersects  $\overline{AB}$ ,  $\overline{AC}$ ,  $\overline{DE}$ , and  $\overline{EF}$  at  $W$ ,  $X$ ,  $Y$ , and  $Z$ , respectively, such that  $\tfrac{AW}{BW} = \tfrac29$  ,  $\tfrac{AX}{CX} = \tfrac56$  , and  $\tfrac{DY}{EY} = \tfrac92$ . The ratio  $\tfrac{EZ}{FZ}$  can then be written as  $\tfrac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .
[asy]
size(200);
defaultpen(linewidth(0.6));
real r = 3/11, s = 0.52, l = 33, d=5.5;
pair A = (l/2,l*sqrt(3)/2), B = origin, C = (l,0), D = (3*l/2+d,l*sqrt(3)/2), E = (l+d,0), F = (2*l+d,0);
pair W = r*B+(1-r)*A, X = s*C+(1-s)*A, Y = extension(W,X,D,E), Z = extension(W,X,E,F);
draw(E--D--F--B--A--C^^W--Z);
dot(" $A$ ",A,N);
dot(" $B$ ",B,S);
dot(" $C$ ",C,S);
dot(" $D$ ",D,N);
dot(" $E$ ",E,S);
dot(" $F$ ",F,S);
dot(" $W$ ",W,0.6*NW);
dot(" $X$ ",X,0.8*NE);
dot(" $Y$ ",Y,dir(100));
dot(" $Z$ ",Z,dir(70));
[/asy]

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