
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 equations:<br/>The first equation: ${a}_{1}=\frac{1}{1×3}=\frac{1}{2}×(1-\frac{1}{3})$;<br/>The second equation: ${a}_{2}=\frac{1}{3×5}=\frac{1}{2}×(\frac{1}{3}-\frac{1}{5})$;<br/>The third equation: ${a}_{3}=\frac{1}{5×7}=\frac{1}{2}×(\frac{1}{5}-\frac{1}{7})$;<br/>$\ldots $<br/>Answer the following questions:<br/>$(1)$ According to the above pattern, write out the fifth equation: $a_{5}=\_\_\_\_\_\_.$<br/>$(2)$ Express the $n$th equation with an algebraic expression: $a_{n}=\_\_\_\_\_\_=\_\_\_\_\_\_(n$ is a positive integer);<br/>$(3)$ Find the value of $a_{1}+a_{2}+\ldots +a_{100}$.

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