
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
For any real number \( x \), let \( \lfloor x \rfloor \) denote the greatest integer less than or equal to \( x \), \( \{ x \} = x - \lfloor x \rfloor \), and \( \lceil x \rceil \) denote the smallest integer greater than or equal to \( x \). If \( x_{1}, x_{2}, \cdots, x_{m} \) \( \left(0 \leq x_{1} < x_{2} < \cdots < x_{m} \leq 10 \right) \) are all the real numbers from 0 to 10 that satisfy the equation \( \lfloor x \rfloor \{ x \} \lceil x \rceil = 1 \), find the value of \( x_{1} + x_{2} + \cdots + x_{m} \).

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