
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 $f(x) = x^2 + bx + c$.
(1) If $f(x)$ is an even function, and $f(1) = 0$, find the maximum and minimum values of the function $f(x)$ in the interval $[-1, 3]$;
(2) To make $f(x)$ a monotonic function in the interval $[-1, 3]$, find the range of values for $b$.

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