
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
Collinear points  $A$ ,  $B$ , and  $C$  are given in the Cartesian plane such that  $A= (a, 0)$  lies along the x-axis,  $B$  lies along the line  $y=x$ ,  $C$  lies along the line  $y=2x$ , and  $\frac{AB}{BC}=2$ . If  $D= (a, a)$ , and the circumcircle of triangle  $ADC$  intersects the line  $y=x$  again at  $E$ , and ray  $AE$  intersects  $y=2x$  at  $F$ , evaluate  $\frac{AE}{EF}$ .

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