
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
The teacher wrote three equations on the blackboard. Please observe carefully, find the pattern, and answer the following questions.<br/>Observe the following equations:①$3^{2}-1^{2}=8\times 1$；②$5^{2}-3^{2}=8\times 2$；③$7^{2}-5^{2}=8\times 3$；$\ldots $<br/>$(1)$Based on the patterns observed in the three equations above, write down the fourth equation:______；<br/>$(2)$Write down your conjecture for the $n$-th equation (expressed in terms of $n$), and verify its correctness.

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