
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 $d$, the sum of its first $n$ terms is $S_n$. If the two intersection points of the line $y=a_1x$ and the circle $(x-2)^2+y^2=1$ are symmetric about the line $x+y+d=0$, determine the formula for the sum $S_n$.

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