
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
Given a natural number \( n \). For any number \( x \), consider the sum \( \mathrm{Q}(x) = \left\lfloor x \right\rfloor + \left\lfloor\frac{\{x\}}{2}\right\rfloor + \left\lfloor\frac{\{x\}}{3}\right\rfloor + \cdots + \left\lfloor\frac{\{x\}}{10 \cdot n}\right\rfloor \). Find the difference \( Q(10^n) - Q(10^n - 1) \). (Here, \( \left\lfloor x \right\rfloor \) denotes the floor function of \( x \), i.e., the greatest integer less than or equal to \( x \)).

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