
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
If $(1-2x)^{2010} = a + a_1x + a_2x^2 + \ldots + a_{2010}x^{2010}$ ($x \in \mathbb{R}$), then the value of $(a+a_1) + (a+a_2) + (a+a_3) + \ldots + (a+a_{2010}) =$ ?

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