
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 $2$, the sum of consecutive even numbers is shown in the table below:

| Number of terms $n$ | $S$       |
|----------------------|-----------|
| $1$                  | $2=1\times 2$ |
| $2$                  | $2+4=6=2\times 3$ |
| $3$                  | $2+4+6=12=3\times 4$ |
| $4$                  | $2+4+6+8=20=4\times 5$ |
| $5$                  | $2+4+6+8+10=30=5\times 6$ |

$(1)$ If $n=8$, then the value of $S$ is ______.<br/>$(2)$ According to the pattern in the table, it is conjectured that the formula for expressing $S$ in terms of $n$ is: $S=2+4+6+8+\ldots +2n=\_\_\_\_\_\_.$<br/>$(3)$ Calculate the value of $102+104+106+\ldots +210+212$ based on the pattern in the previous question.

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