
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 non-decreasing sequence of odd integers \(\left(a_{1}, a_{2}, a_{3}, \dots\right)=(1,3,3,5,5,5,5,5, \dots)\), each positive odd number \(k\) appears \(k\) times. Given that there exist integers \(b\), \(c\), and \(d\) such that for all positive integers \(n\), \(a_{n}=b[\sqrt{n+c}]+d\), find \(b\), \(c\), and \(d\).

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