
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 $|k - \sqrt{p}| < \frac{1}{2}$. Find $T = \sum_{p=1}^{1000} b(p)$.

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