
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 the diagram below, triangle $ABC$ has been reflected over its altitude $\overline{AH}$ to produce triangle $AB'C'$. If $AE = 8$, $EF = 4$, $FC = 8$, and $BD = 15$, then find $AB$.

[asy]
size(250);
pair A,B,C,H,BB,CC,E,F;
B = (0,0);
D = (15,0);
H = (7.5,0);
C = 2*H;
A = D + (scale(1.2)*rotate(aCos((225-144-25)/120))*(H-D));
CC = D + D + D - A - A;
BB = reflect(A,H)*B;
E = A + 0.4*(D-A);
F = A + 0.8*(D-A);
draw(H--A--BB--CC--A--B--C--A);
label("$H$", H, S);
label("$A$", A, NE);
label("$B$", B, SW);
label("$C$", C, SE);
label("$C'$", CC, S);
label("$B'$", BB, E);
label("$D$", D, NW);
label("$E$", E, W);
label("$F$", F, W);
label("$4$", (E+F)/2, N);
label("$8$", (A+F)/2, S);
label("$8$", (A+E)/2, S);
label("$15$", D/2, S);
[/asy]

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