
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 $f(x)=ax^{3}+bx^{2}-x$ ($x\in\mathbb{R}, a, b$ are constants, $a\neq 0$), and when $x=1$ and $x=2$, the function $f(x)$ takes extreme values.  
$(1)$ Find the expression of $f(x)$;  
$(2)$ If the curve $y=f(x)$ and $g(x)=-3x-m$ ($-2\leqslant x\leqslant 0$) have two different intersection points, find the range of the real number $m$.

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