
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 the first $n$ terms denoted by $S_n$, if the line $y=a_1x+m$ is perpendicular to the line $x+2y-d=0$ whose y-intercept is $1$, then the sum of the first $100$ terms of the sequence $\left\{ \frac{1}{S_n} \right\}$ is __________.

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