
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 triangle $ABC$, two lines $BD$ and $BE$ meet side $AC$ at points $D$ and $E$ respectively, such that $BD$ segments $\angle ABC$ in the ratio 1:2 (i.e., $\angle ABD : \angle DBC = 1:2$) and $BE$ bisects $\angle DBC$. Determine the ratio $\frac{AD}{EC}$.
**A)** $\frac{AB \cdot BC}{BD \cdot BC}$ 
**B)** $\frac{AB \cdot BC}{2BD \cdot BC}$
**C)** $\frac{AB \cdot (BC - BD)}{2BD \cdot BC}$
**D)** $\frac{AB \cdot (BC - BD)}{BD \cdot BC}$ 
**E)** $\frac{AB \cdot (BC - BD)}{BD(BC - BD)}$

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