
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 $f(x)=ax^{3}+bx^{2}+cx$ has extreme values at $x=±1$, and the slope of the tangent line at $x=0$ is $-3$.
$(1)$ Find the expression of $f(x)$;
$(2)$ Find the equation of the tangent line passing through point $A(2,2)$.

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