$ABCD$ 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]
1. **Identify the dimensions and relationships:**
   - Given: \( AE = 8 \), \( BE = 17 \), and \( CF = 3 \).
   - \( AB = AE + EB = 8 + 17 = 25 \).

2. **Determine the coordinates of points:**
   - Let \( A = (0, 0) \), \( B = (25, 0) \), \( D = (0, h) \), and \( C = (25, h) \).
   - \( E = (8, 0) \) and \( F = (x, h) \) where \( x \) is to be determined.

3. **Use the Pythagorean theorem to find \( AB' \):**
   - Since \( B \) is folded to \( B' \) on \( AD \), the distance \( AB' \) is the same as the distance from \( B \) to the line \( AD \).
   - \( AB' = \sqrt{AE^2 + BE^2} = \sqrt{8^2 + 17^2} = \sqrt{64 + 289} = \sqrt{353} = 15 \).

4. **Determine the coordinates of \( B' \):**
   - Since \( B' \) lies on \( AD \), \( B' \) has coordinates \( (0, y) \).
   - The distance from \( B \) to \( B' \) is \( 15 \), so \( y = 15 \).

5. **Determine the coordinates of \( F \):**
   - Given \( CF = 3 \), and \( C = (25, h) \), \( F = (x, h) \) where \( x \) is to be determined.
   - Since \( F \) lies on \( CD \), \( F \) has coordinates \( (x, h) \).

6. **Use similar triangles to find \( x \):**
   - Triangles \( AB'E \) and \( C'KF \) are similar.
   - \( \frac{C'F}{AE} = \frac{B'E}{AB} \).
   - \( \frac{3}{8} = \frac{15}{25} \).
   - Solving for \( x \), we get \( x = \frac{3 \cdot 25}{15} = 5 \).

7. **Calculate the perimeter of rectangle \( ABCD \):**
   - The width \( AB = 25 \).
   - The height \( AD = h \).
   - Since \( F \) is on \( CD \), \( h = 3 \).
   - The perimeter \( P = 2(AB + AD) = 2(25 + 3) = 2 \cdot 28 = 56 \).

8. **Express the perimeter in the form \( \frac{m}{n} \):**
   - The perimeter \( P = 56 \) can be written as \( \frac{56}{1} \).

9. **Find \( m + n \):**
   - Here, \( m = 56 \) and \( n = 1 \).
   - Therefore, \( m + n = 56 + 1 = 57 \).

The final answer is \( \boxed{57} \).