
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:
$$(x+2)^8 = a_0 + a_1(x+1) + a_2(x+1)^2 + \dots + a_8(x+1)^8,$$
where $a_i$ ($i=0,1,\dots,8$) are real constants, find the value of $a_1+2a_2+ \dots +7a_7+8a_8$.

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