
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
Find the sum of the $1004$ roots of $(x-1)^{1004} + 2(x-2)^{1003} + 3(x-3)^{1002} + \cdots + 1003(x-1003)^2 + 1004(x-1004)$.

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