
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
Observing the following expressions: $1\times 3+1=2^{2}$, $2\times 4+1=3^{2}$, $3\times 5+1=4^{2}$, $4\times 6+1=5^{2}$, $\ldots $,  
$(1)$ Please write out the $6$th expression according to the above rule: ______;  
$(2)$ Please write out the $n$th expression: ______;  
$(3)$ Calculate: $(1+\frac{1}{1\times 3})\times (1+\frac{1}{2\times 4})\times (1+\frac{1}{3\times 5})\times \ldots \times (1+\frac{1}{98\times 100})$.

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