
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 two functions $f(x)=8x^2+16x-k$ and $g(x)=2x^3+5x^2+4x$, where $k$ is a real number.  
(1) For any $x \in [-3,3]$, $f(x) \leq g(x)$ holds, find the range of values for $k$.  
(2) There exists $x \in [-3,3]$ such that $f(x) \leq g(x)$ holds, find the range of values for $k$.  
(3) For any $x_1, x_2 \in [-3,3]$, $f(x_1) \leq g(x_2)$ holds, find the range of values for $k$.

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