
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
Find the distance  $\overline{CF}$  in the diagram below where  $ABDE$  is a square and angles and lengths are as given:
[asy]
markscalefactor=0.15;
size(8cm);
pair A = (0,0);
pair B = (17,0);
pair E = (0,17);
pair D = (17,17);
pair F = (-120/17,225/17);
pair C = (17+120/17, 64/17);
draw(A--B--D--E--cycle^^E--F--A--cycle^^D--C--B--cycle);  
label(" $A$ ", A, S);
label(" $B$ ", B, S);
label(" $C$ ", C, dir(0));
label(" $D$ ", D, N);
label(" $E$ ", E, N);
label(" $F$ ", F, W);
label(" $8$ ", (F+E)/2, NW);
label(" $15$ ", (F+A)/2, SW);
label(" $8$ ", (C+B)/2, SE);
label(" $15$ ", (D+C)/2, NE);
draw(rightanglemark(E,F,A));
draw(rightanglemark(D,C,B));
[/asy]
The length  $\overline{CF}$  is of the form  $a\sqrt{b}$  for integers  $a, b$  such that no integer square greater than  $1$  divides  $b$ . What is  $a + b$ ?

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