
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 the following series of equations: $1×\frac{1}{2}=1-\frac{1}{2}$; $\frac{1}{2}×\frac{1}{3}=\frac{1}{2}-\frac{1}{3}$; $\frac{1}{3}×\frac{1}{4}=\frac{1}{3}-\frac{1}{4}$; $\frac{1}{4}×\frac{1}{5}=\frac{1}{4}-\frac{1}{5}⋅⋅⋅⋅⋅⋅$<br/>$(1)$ Please express the pattern you discovered with an equation containing $n$______;<br/>$(2)$ Use the equations to calculate: $\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)}+\frac{1}{(x+3)(x+4)}$;<br/>$(3)$ Calculate: $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+2}+⋅⋅⋅+\frac{1}{3+\sqrt{10}}$.

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