
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 $x_1+1 = x_2+2 = x_3+3 = \cdots = x_{1000}+1000 = x_1 + x_2 + x_3 + \cdots + x_{1000} + 1001$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1000} x_n$.

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