
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 $f(x) = x^3 + 3ax^2 - 9x + 5$, if $f(x)$ has an extreme value at $x = 1$.

(1) Find the value of the real number $a$;

(2) Find the extreme value of the function $f(x)$;

(3) If for any $x \in [-4, 4]$, there is $f(x) < c^2$, find the range of values for the real number $c$.

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