
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 \( x_1+1 = x_2+2 = x_3+3 = \cdots = x_{10}+10 = x_1+x_2+x_3+\cdots+x_{10}+11 \). Find the value of \( \left\lfloor |S| \right\rfloor \), where \( S = \sum_{n=1}^{10} x_n \).

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