
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 \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[ 
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots 
\]

A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy 
\[ 
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0 
\]

Find the sum of the lengths of all intervals on the number line that represent all such "good numbers".

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