
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 and answer the questions:<br/>The first equation: $2^{2}-1^{2}=3$;<br/>The second equation: $3^{2}-2^{2}=5$;<br/>The third equation: $4^{2}-3^{2}=7$;<br/>$\ldots $<br/>The $n$th equation: ______. $(n$ is an integer, and $n\geqslant 1)$<br/>【Attempt】<br/>$(1)$ According to the above pattern, write down the fourth equation: ______;<br/>【Discovery】<br/>$(2)$ Based on this pattern, write down your conjectured $n$th equation and explain its correctness;<br/>【Application】<br/>$(3)$ Using the above pattern, directly calculate the value of $2023^{2}-2022^{2}$ as ______.<br/>$(4)$ Using the above pattern, find the value of $3+5+7+\ldots +197$.

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