
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 a sequence of positive numbers \( \{a_n\} \) such that \( a_1 = 1 \) and \( S_n = \frac{1}{2} \left( a_n + \frac{1}{a_n} \right) \), where \( S_n \) is the sum of the first \( n \) terms of the sequence \( \{a_n\} \), find \( \left[ \frac{1}{S_1} + \frac{1}{S_2} + \cdots + \frac{1}{S_{100}} \right] \).

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