
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
Explore and discover: $\frac{1}{1×2}=1-\frac{1}{2}$; $\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$; $\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}\ldots $<br/>Based on the pattern you discovered, answer the following questions:<br/>$(1)$ $\frac{1}{4×5}=$______,$\frac{1}{n×(n+1)}=$______; <br/>$(2)$ Calculate using the pattern you discovered: $\frac{1}{1×2}+\frac{1}{2×3}+\frac{1}{3×4}+\ldots +\frac{1}{n×(n+1)}$<br/>$(3)$ Flexibly use the pattern to solve the equation: $\frac{1}{x(x+2)}+\frac{1}{(x+2)(x+4)}+\ldots +\frac{1}{(x+98)(x+100)}=\frac{1}{x+100}$.

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