
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
Let $\alpha$ and $\beta$ be the real roots of the equation \(x^{2}-x-2021=0\), where $\alpha > \beta$. Define

\[ A = \alpha^{2} - 2 \beta^{2} + 2 \alpha \beta + 3 \beta + 7 \]

Find the largest integer not exceeding \(A\).

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