
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
Observe the following expressions:<br/>$\frac{1}{6}=\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$,$\frac{1}{12}=\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$,$\frac{1}{20}=\frac{1}{4×5}=\frac{1}{4}-\frac{1}{5}$,$\ldots \ldots $<br/>$(1)$ It can be inferred that $\frac{1}{42}=$______；<br/>$(2)$ Please express the general rule of (1) using an equation containing the letter $m$ ($m$ is a positive integer)______；<br/>$(3)$ Solve the equation using the method above: $\frac{1}{(x-1)(x-2)}+\frac{1}{x(x-1)}=\frac{1}{x}$.

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