
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 $f(x)=x^{3}+x-16$.  
$(1)$ Find the equation of the tangent line to the curve $y=f(x)$ at the point $(2,6)$.  
$(2)$ If line $l$ is the tangent to the curve $y=f(x)$ and passes through the origin, find the equation of line $l$ and the coordinates of the tangent point.

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