
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 $f(x)=x^{3}-2x+2$.  
$(1)$ Find the average rate of change of the function $f(x)$ on the interval $[0,2]$.  
$(2)$ Let $g(x)=2x+\frac{k}{x}$. If the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is parallel to the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$, find the value of the real number $k$.  
$(3)$ Find the equation of the line passing through the point $(2,f(2))$ and tangent to the curve $y=f(x)$.

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