
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 $x_1+3=x_2+4=x_3+5=\cdots=x_{1004}+1006=x_1+x_2+x_3+\cdots+x_{1004}+1005$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1004}x_n$.

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