
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 $\triangle ABC$, segments $AD$ and $AE$ bisect $\angle BAC$ such that $AD$ divides $\angle BAC$ into two angles in the ratio $2:3$ and $AE$ into $1:1$. The lengths of $BD$, $DE$, and $EC$ are $4$, $2$, and $5$, respectively. Determine the length of the shortest side of $\triangle ABC$.
A) $\frac{30}{11}$  
B) $\frac{36}{11}$  
C) $6$  
D) $5$  
E) not uniquely determined by the given information

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