
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
Observe the following equations:
$$
\begin{matrix}
(1+x+x^{2})^{1}=1+x+x^{2}, \\
(1+x+x^{2})^{2}=1+2x+3x^{2}+2x^{3}+x^{4}, \\
(1+x+x^{2})^{3}=1+3x+6x^{2}+7x^{3}+6x^{4}+3x^{5}+x^{6}, \\
(1+x+x^{2})^{4}=1+4x+10x^{2}+16x^{3}+19x^{4}+16x^{5}+10x^{6}+4x^{7}+x^{8}, \\
\ldots
\end{matrix}
$$
Based on the above equations, for $n \in \mathbb{N}$, if $(1+x+x^{2})^{n}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{2n}x^{2n}$, then $a_{2}$ equals $\_\_\_\_\_\_$.

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