
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
Given the sequence $\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}$, where each odd positive integer $k$ appears $k$ times, find the sum of the integers $b$, $c$, and $d$ such that for all positive integers $n$, $a_n=b\lfloor \sqrt{n+c} \rfloor +d$.

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