
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  
$$(1+x+x^{2})^{1} = 1+x+x^{2},$$  
$$(1+x+x^{2})^{2} = 1+2x+3x^{2}+2x^{3}+x^{4},$$  
$$(1+x+x^{2})^{3} = 1+3x+6x^{2}+7x^{3}+6x^{4}+3x^{5}+x^{6},$$  
$$(1+x+x^{2})^{4} = 1+4x+10x^{2}+16x^{3}+19x^{4}+16x^{5}+10x^{6}+4x^{7}+x^{8},$$  
...  
If $(1+x+x^{2})^{6} = a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{12}x^{12}$, then find $a_{2}$.

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