
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
 $ABCD$  is a rectangular sheet of paper that has been folded so that corner  $B$  is matched with point  $B'$  on edge  $AD$ . The crease is  $EF$ , where  $E$  is on  $AB$  and  $F$ is on  $CD$ . The dimensions  $AE=8$ ,  $BE=17$ , and  $CF=3$  are given. The perimeter of rectangle  $ABCD$  is  $m/n$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C;
draw(F--D--A--E);
draw(E--B--C--F, linetype("4 4"));
filldraw(E--F--Cp--Bp--cycle, white, black);
pair point=( 12.5, 35/3 );
label(" $A$ ", A, dir(point--A));
label(" $B$ ", B, dir(point--B));
label(" $C$ ", C, dir(point--C));
label(" $D$ ", D, dir(point--D));
label(" $E$ ", E, dir(point--E));
label(" $F$ ", F, dir(point--F));
label(" $B^\prime$ ", Bp, dir(point--Bp));
label(" $C^\prime$ ", Cp, dir(point--Cp));[/asy]

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