
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 \( p \), let \( b(p) \) denote the unique positive integer \( k \) such that \( |k - \sqrt{p}| < \frac{1}{2} \). Calculate \( S = \sum_{p=1}^{3000} b(p) \).

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