
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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