
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 transformation pattern:<br/>$\frac{1}{1×3}=\frac{1}{2}×(1-\frac{1}{3})$;<br/>$\frac{1}{3×5}=\frac{1}{2}×(\frac{1}{3}-\frac{1}{5})$;<br/>$\frac{1}{5×7}=\frac{1}{2}×(\frac{1}{5}-\frac{1}{7})$;<br/>$\ldots $<br/>Answer the following questions:<br/>$(1)$ The $5$th expression is ______;<br/>$(2)$ If $n$ is a positive odd integer, guess $\frac{1}{n(n+2)}=$______;<br/>Based on your insight, try to answer the following question: If rational numbers $a$ and $b$ satisfy $|a-1|+\left(b-3\right)^{2}=0$, find the value of $\frac{1}{ab}+\frac{1}{(a+2)(b+2)}+\frac{1}{(a+4)(b+4)}+…+\frac{1}{(a+100)(b+100)}$.

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