
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 a desert, the oases $A$, $B$, $C$, and $D$ are located at the vertices of a rhombus with a $60^\circ$ angle. $B$ and $D$ are at the endpoints of the longer diagonal, and the side of the rhombus is $15 \text{ km}$ long. An optimal road system is to be built that connects any two oases. Ali suggests connecting $A$ directly to $B$, $C$, and $D$. Hasszán suggests building the two diagonals, claiming they are shorter and fairer. Ibrahim finally provides the shortest total length road system. This system is built, and the costs are distributed among the four oases so that each oasis's contribution is proportional to the total distance traveled by one of its residents if they visit each of the other oases exactly once. What is the cost distribution ratio?

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