
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 first equation: $\frac{1}{2}=\frac{{1+3×2}}{{2+3×4}}$;<br/>The second equation: $\frac{2}{3}=\frac{{2+3×4}}{{3+3×6}}$;<br/>The third equation: $\frac{3}{4}=\frac{{3+3×6}}{{4+3×8}}$;<br/>$\cdots $<br/>Based on the information provided by the above three equations, solve the following problems.<br/>$(1)$ Write down the fourth equation: ______;<br/>$(2)$ Find the pattern according to the above equations and make a conjecture for the $n$-th equation, and provide a proof.

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