
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 \( D = (a, a) \), the circumcircle of triangle \( ADC \) intersects \( y = x \) 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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