
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 $a+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{12}(x+12)^{12}=(x^{2}-2x-2)^{6}$, where $a_{i}$ are constants. Find the value of $2a_{2}+6a_{3}+12a_{4}+20a_{5}+\ldots+132a_{12}$.

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