
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
Starting from $2$, consecutive even numbers are added together. Observe the following equations:<br/>$2=1^{2}+1$.<br/>$2+4=2^{2}+2$.<br/>$2+4+6=3^{2}+3$.<br/>$2+4+6+8=4^{2}+4$.<br/>$\ldots $<br/>According to the pattern, answer the following questions:<br/>$(1)$ Write down the $5$th equation: ______；<br/>$(2)$ ① Write down the $n$th equation: ______; (express using $n$)<br/>② Calculate: $102+104+106+\ldots +198+200$.

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