
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 functions $f(x)=2x^{2}+x-k$ and $g(x)=ax^{3}+bx^{2}+cx+d(a\neq 0)$ are odd functions on $R$, where $g(x)$ achieves a minimum value of $-2$ when $x=1$. Find:<br/>
$(1)$ The monotonic intervals and maximum value of function $g(x)$;<br/>
$(2)$ The range of real number $k$ if $f(x)\leqslant g(x)$ for all $x\in [-1,3]$;<br/>
$(3)$ The range of real number $k$ if $f(x_{1})\leqslant g(x_{2})$ for all $x_{1}\in [-1,3]$ and $x_{2}\in [-1,3]$.

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