
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 equation $x^2 + bx + 1 = 0$ has two real roots $\alpha$ and $\beta$:

(1) Can the relationship between $\alpha^2 + \beta^2$ and 2 be determined? Please explain your reasoning.
(2) If $\alpha > \beta$, and the triangle with sides $\alpha^2 + \beta^2$, $3\alpha - 3\beta$, and $\alpha\beta$ is an isosceles triangle, find the value of $b$.

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