
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
Given an arithmetic sequence $\{a_n\}$ with the first term $a_1$ and common difference $d$, and the sum of its first $n$ terms is $S_n$. If the line $y = a_1x$ and the circle $(x - 2)^2 + y^2 = 4$ have two intersection points that are symmetric with respect to the line $x + y + d = 0$, then $S_n = \_\_\_\_\_\_\_\_$.

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