
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 $f(x)=kx^{3}+3(k-1)x^{2}-k^{2}+1$ that attains extreme values at $x=0,x=4$.

(I) Find the value of the constant $k$;
(II) Let $g(x)=f(x)+c$, and $\forall x\in [-1,2]$, $g(x)\geqslant 2c+1$ always holds true. Find the range of $c$.

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