
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 $f(x)=4x^{2}+\frac{1}{x}-a$, $g(x)=f(x)+b$, where $a$, $b$ are constants.

$(1)$ If $x=1$ is an extremum point of the function $y=xf(x)$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;

$(2)$ If the function $f(x)$ has $2$ zeros, and $f(g(x))$ has $6$ zeros, find the range of values for $a+b$.

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