
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/>$(1)1^{2}=2^{2}-3$;<br/>$(2)2^{2}=3^{2}-5$;<br/>$(3)3^{2}=4^{2}-7$;<br/>$(4)4^{2}=5^{2}-9$;<br/>$\ldots $.<br/>According to the above pattern, answer the following questions:<br/>$(1)$ Write down the equation containing $8^{2}$: ______;<br/>$(2)$ Write down the $n$th equation: ______(expressed using an algebraic expression containing $n$);<br/>$(3)$ When $n\geqslant 2$, calculate: $3+5+7+9+\ldots +(2n-1)$.

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