
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
In circle  $\Omega$ , let  $\overline{AB}=65$  be the diameter and let points  $C$  and  $D$  lie on the same side of arc  $\overarc{AB}$  such that  $CD=16$ , with  $C$  closer to  $B$  and  $D$  closer to  $A$ . Moreover, let  $AD, BC, AC,$  and  $BD$  all have integer lengths. Two other circles, circles  $\omega_1$  and  $\omega_2$ , have  $\overline{AC}$  and  $\overline{BD}$  as their diameters, respectively. Let circle  $\omega_1$  intersect  $AB$  at a point  $E \neq A$  and let circle  $\omega_2$  intersect  $AB$  at a point  $F \neq B$ . Then  $EF=\frac{m}{n}$ , for relatively prime integers  $m$  and  $n$ . Find  $m+n$ .
[asy]
size(7cm);
pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, NE);
dot(" $D$ ", D, NW);
dot(" $E$ ", E, S);
dot(" $F$ ", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label(" $\mathcal P$ ", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label(" $\mathcal Q$ ", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label(" $\mathcal R$ ", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
*Proposed by **AOPS12142015***

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