
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
Define: If for some $x_0$ in the domain of a function $f(x)$ it holds that $f(x_0) = x_0$, then $x_0$ is called a fixed point of $f(x)$. Given the function $f(x) = ax^2 + (b + 1)x + b - 1$ with $a \neq 0$,
(1) Find the fixed points of function $f(x)$ when $a = 1$ and $b = 3$;
(2) If for any real number $b$, the function $f(x)$ always has two fixed points, find the range of values for $a$;
(3) Under the condition of (2), if the abscissas of two points A and B on the graph of $y = f(x)$ are the fixed points of the function $f(x)$, and the midpoint C of A and B lies on the graph of $g(x) = -x + \frac{2a}{5a^2 - 4a + 1}$, find the minimum value of $b$.
(Refer to the formula: The coordinates of the midpoint for points A$(x_1, y_1)$ and B$(x_2, y_2)$ are $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.)

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