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

1. When \(a = 1\) and \(b = -2\), find the fixed points of the function \(f(x)\).
2. If for any real number \(b\), the function \(f(x)\) always has two distinct fixed points, find the range of values for \(a\).
3. Under the condition in (2), if the points \(A\) and \(B\) on the graph of \(y = f(x)\) have abscissas which are fixed points of \(f(x)\) and the points \(A\) and \(B\) are symmetric with respect to the line \(y = kx + \frac{1}{2a^2 + 1}\), find the minimum value of \(b\).

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