
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
Let the function $f(x)=x^{3}+3ax^{2}-9x+5$. If $f(x)$ has an extremum at $x=1$.

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

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

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