
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 a quadratic function $f(x) = ax^2 + bx + c$, where the constants $a$, $b$, and $c \in \mathbb{R}$.
(1) If $f(3) = f(-1) = -5$ and the maximum value of $f(x)$ is $3$, find the explicit expression for the function $f(x)$.
(2) For $a = 1$, if for any $x_1, x_2 \in [-1, 1]$, it holds that $|f(x_1) - f(x_2)| \leq 4$, find the range of possible values for $b$.

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