
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
Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\cdots=x_{1500}+3000=\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{1500}x_n$.

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