
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 $(x-1)^4(x+2)^8 = ax^{12} + a_1x^{11} + \ldots + a_{n}x + a_{12}$, find the value of $a_2 + a_4 + \ldots + a_{12}$.

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