
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
Given that $\triangle ABC$ is an equilateral triangle, with $M$ and $N$ as the midpoints of sides $BC$ and $BA$ respectively, and $O$ is the circumcenter of $\triangle BMN$. Let $D$ be a point on side $AC$ such that $S_{\triangle AOD} = \frac{1}{n} S_{\triangle ABC}$, with $n > 1$ and $n \in \mathbf{N}_{+}$. Determine the value of $\frac{AD}{DC}$.


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