
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: $\frac{1}{6}=\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$, $\frac{1}{12}=\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$, $\frac{1}{20}=\frac{1}{4×5}=\frac{1}{4}-\frac{1}{5}$, $\frac{1}{30}=\frac{1}{5×6}=\frac{1}{5}-\frac{1}{6},⋅⋅⋅$  
$(1)$ It can be inferred from this that $\frac{1}{42}=$______;  
$(2)$ Following the above pattern, write down the process for $\frac{1}{240}$;  
$(3)$ Please come up with a general rule that can represent the above expressions, express it as an equation involving $m$ (where $m$ represents an integer), and explain the reasoning;  
$(4)$ Directly calculate the value of $\frac{1}{(x-2)(x-3)}-\frac{2}{(x-1)(x-3)}+\frac{1}{(x-1)(x-2)}$ using the rule from $(3)$.

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