
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
In the modified sequence of odd integers $\{b_1, b_2, b_3, \ldots \} = \{1, 1, 1, 3, 3, 3, 3, 3, 5, 5, \ldots\}$, where each odd positive integer $k$ appears $k+2$ times, it is a fact that there are integers $p, q$, and $r$ such that for all positive integers $n$, $b_n = p \lfloor \sqrt{n+q} \rfloor + r$, where $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. The sum $p+q+r$ equals how much?
A) 0
B) 1
C) 2
D) 3
E) 4

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