
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 first term $a\_1$ and common difference $d$, let $S\_n$ denote the sum of its first $n$ terms. If the two intersection points of the line $y=a\_1x+m$ and the circle $x^2+(y-1)^2=1$ are symmetric about the line $x+y-d=0$, find the sum of the first 100 terms of the sequence $\{(\frac{1}{S\_n})\}$.

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