
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, $\triangle ABF$, $\triangle BCF$, and $\triangle CDF$ are right-angled triangles, with $\angle ABF = 45^\circ$, $\angle BCF = 90^\circ$, and $\angle CDF = 45^\circ$. Let $AB = 20$. Find the length of segment $CF$.
[asy]
pair A, B, C, D, F;
A=(0,14.14);
B=(0,0);
C=(10,-10);
D=(15,-15);
F=(10,0);
draw(A--B--C--D--F--A);
draw(B--F);
draw(C--F);
label("A", A, N);
label("B", B, W);
label("C", C, SW);
label("D", D, S);
label("F", F, NE);
[/asy]

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