
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 functions $f(x)=8x^{2}+16x-k$ and $g(x)=2x^{3}+5x^{2}+4x$, where $k$ is a real number.

    $(1)$ For any $x \in [-3,3]$, if $f(x) \leqslant g(x)$ always holds, find the range of $k$;

    $(2)$ If there exists $x \in [-3,3]$ such that $f(x) \leqslant g(x)$ holds, find the range of $k$;

    $(3)$ For any $x_{1}, x_{2} \in [-3,3]$, if $f(x_{1}) \leqslant g(x_{2})$ always holds, find the range of $k$.

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