
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  $|k-\sqrt{p}|<\frac{1}{2}$ .  For example,  $b(6) = 2$  and  $b(23)=5$ .  If  $S = \textstyle\sum_{p=1}^{2007}b(p)$ , find the remainder when S is divided by 1000.

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