
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 function $f(x)$, if there exists an $x \in \mathbb{R}$ such that $f(x_0) = x_0$, then $x_0$ is called a fixed point of $f(x)$. Consider the function $f(x) = ax^2 + (b + 1)x + (b - 1) (a \neq 0)$.

1. Find the fixed point(s) of the function $f(x)$ when $a = 1$ and $b = 2$.
2. Determine the range of values for $a$ such that the function $f(x)$ always has two distinct fixed points for any real number $b$.
3. Under the conditions of part 2, if the two fixed points of $f(x)$ are $x_1$ and $x_2$, and $f(x_1) + x_2 = \frac{-a}{2a^2 + 1}$, find the range of values for the real number $b$.

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