
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 non-decreasing odd number sequence {a_{1}, a_{2}, a_{3}, \cdots} = {1, 3, 3, 3, 5, 5, 5, 5, 5, \cdots}, where each positive odd number k appears k times, find the value of b + c + d in the equation a_{n} = b \left\lfloor \sqrt{n+c} \right\rfloor + d for all positive integers n.

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