
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 $\{a_1, a_2, a_3, \ldots\}$ where each positive even integer $k$ appears exactly $k+1$ times, and the sequence starts with 2, determine the sum of the constants $b$, $\alpha$, $\beta$, and $d$ in the formula $a_n = b\lfloor \alpha\sqrt{n} + \beta \rfloor + d$ such that the sequence retains its noted pattern for all positive integers $n$.

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