
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 an arbitrary number \( x \), consider the sum 

$$
Q(x) = \left\lfloor \frac{x}{1} \right\rfloor + \left\lfloor \frac{\{x\}}{2} \right\rfloor + \left\lfloor \frac{\{x\}}{3} \right\rfloor + \left\lfloor \frac{\{x\}}{4} \right\rfloor + \cdots + \left\lfloor \frac{\{x\}}{10^n} \right\rfloor.
$$

Find the difference 

$$
Q\left(10^n\right) - Q\left(10^n - 1\right),
$$

where \( \left\lfloor x \right\rfloor \) denotes the integer part of \( x \), that is, the largest integer not exceeding \( x \).

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