
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: $\frac{1}{1+2}+\frac{2}{{1}^{2}+2×1}=\frac{1}{1}$;<br/>The $2$nd equation: $\frac{1}{2+2}+\frac{2}{{2}^{2}+2×2}=\frac{1}{2}$;<br/>The $3$rd equation: $\frac{1}{3+2}+\frac{2}{{3}^{2}+2×3}=\frac{1}{3}$;<br/>The $4$th equation: $\frac{1}{4+2}+\frac{2}{{4}^{2}+2×4}=\frac{1}{4}$;<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 (expressed in terms of $n$, where $n$ is a positive integer), and explain why the equation holds.

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