
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
For each positive integer \(1 \leqq k \leqq 100\), let \(a_{k}\) denote the sum \(\frac{1}{k}+\frac{1}{k+1}+\ldots+\frac{1}{100}\). Calculate the value of

\[ a_{1} + a_{1}^{2} + a_{2}^{2} + \ldots + a_{100}^{2}. \]

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