
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
Let the set of inequalities $\begin{cases} x＞0 \\ y＞0 \\ y≤-nx+4n \end{cases}$ ($n\in\mathbb{N}^*$) define a plane region $D_n$. Calculate the value of $\frac{1}{2012}(a_2+a_4+a_6+\ldots+a_{2012})$, where $a_n$ represents the number of lattice points within the region $D_n$.

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