
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 median $\overline{AM}$ to produce triangle $AB'C'$. If $AF = 8$, $FC =16$, and $BE = 12$, then find $AB$.

[asy]
size(250);
pair A,B,C,E,M,BB,CC,FF;
B = (0,0);
E = (12,0);
M = (18,0);
C=2*M;
A = E + (scale(1.2)*rotate(aCos((225-144-25)/120))*(M-E));
CC = E + E + E - A - A;
BB = reflect(A,M)*B;
FF = reflect(A,M)*E;
draw(M--A--BB--CC--A--B--C--A);
label("$M$",M,SE);
label("$A$",A,N);
label("$B$",B,SW);
label("$C$",C,SE);
label("$C'$",CC,S);
label("$B'$",BB,E);
label("$E$",E,NW);
label("$F$",FF,N);
label("$16$",(FF+C)/2,N);
label("$8$",(A+FF)/2,S);
label("$12$",E/2,S);
[/asy]

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