
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
Suppose that \(x_1+1=x_2+2=x_3+3=\cdots=x_{2010}+2010=x_1+x_2+x_3+\cdots+x_{2010}+2011\). Find the value of \(\left\lfloor|T|\right\rfloor\), where \(T=\sum_{n=1}^{2010}x_n\).

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