
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: $\frac{2}{3}-\frac{1}{{1×2×3}}=\frac{1}{2}$;<br/>The $2$nd equation: $\frac{3}{8}-\frac{1}{{2×3×4}}=\frac{1}{3}$;<br/>The $3$rd equation: $\frac{4}{{15}}-\frac{1}{{3×4×5}}=\frac{1}{4}$;<br/>The $4$th equation: $\frac{5}{{24}}-\frac{1}{{4×5×6}}=\frac{1}{5}$;<br/>$\ldots $<br/>Following the above pattern, solve the following problems:<br/>$(1)$ Write down the $6$th equation:______;<br/>$(2)$ Write down your conjectured $n(n$ is a positive integer)th equation:______(express the equation with $n$), and verify the correctness of the equation.

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