
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 a fixed point of $f(x)$. Given the function $f(x) = ax^2 + (b + 1)x + b - 1$ ($a \neq 0$).

(1) When $a = 1$, $b = -2$, find the fixed points of $f(x)$;

(2) If for any real number $b$, the function $f(x)$ always has two distinct fixed points, find the range of values for $a$.

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