
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
Points $B$ and $C$ lie on line segment $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $4$ times the length of $\overline{CD}$. If $D$ lies between $A$ and $C$ and the order of the points, from left to right, is $A, B, D, C$, what fraction of the length of $\overline{AD}$ is $\overline{BC}$?
**A)** $\frac{1}{3}$
**B)** $\frac{2}{3}$
**C)** $\frac{4}{5}$
**D)** $\frac{5}{6}$
**E)** $\frac{1}{2}$

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