
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 denoted by S_n. If the line y = a_1x + m has two intersections with the circle (x-2)^2 + y^2 = 1 that are symmetric with respect to the line x + y + d = 0, find the value of S_n.

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