
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 the sequences of variables $x_1, x_2, \ldots, x_{1005}$ satisfy the equalities $x_1+1 = x_2+3 = x_3+6 = \cdots = x_{1005} + \frac{1005 \cdot 1006}{2} = x_1 + x_2 + \ldots + x_{1005} + 1006$. Find the value of $\left\lfloor |S| \right\rfloor$, where $S = \sum_{n=1}^{1005} x_n$.

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