
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/>${a}_{1}=\frac{1}{1×2}=\frac{1}{1}-\frac{1}{2}$; ${b}_{1}=\frac{1}{1×3}=\frac{1}{2}×(\frac{1}{1}-\frac{1}{3})$;<br/>${a}_{2}=\frac{1}{2×3}=\frac{1}{2}-\frac{1}{3}$; ${b}_{2}=\frac{1}{2×4}=\frac{1}{2}×(\frac{1}{2}-\frac{1}{4})$;<br/>${a}_{3}=\frac{1}{3×4}=\frac{1}{3}-\frac{1}{4}$; ${b}_{3}=\frac{1}{3×5}=\frac{1}{2}×(\frac{1}{3}-\frac{1}{5})$;<br/>Please answer the following questions:<br/>$(1)$ Find the value of $a_{1}+a_{2}+a_{3}+\cdot \cdot \cdot +a_{199}$.<br/>$(2)$ Find the value of $b_{1}+b_{2}+b_{3}+\cdot \cdot \cdot +b_{29}$.

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