
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 $1$st equation: $1-\frac{1}{2^2}=\frac{1}{2}×\frac{3}{2}$;<br/>The $2$nd equation: $1-\frac{1}{3^2}=\frac{2}{3}×\frac{4}{3}$;<br/>The $3$rd equation: $1-\frac{1}{4^2}=\frac{3}{4}×\frac{5}{4}$;<br/>The $4$th equation: $1-\frac{1}{5^2}=\frac{4}{5}×\frac{6}{5}$;<br/>$\ldots \ldots $<br/>According to the above pattern, solve the following problems:<br/>$(1)$ Write down the $6$th equation: ______;<br/>$(2)$ Write down your conjectured $n$th equation: ______;(express the equation with $n$), and explain the reason.

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