
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 $f(x)=x^{3}+ax^{2}+bx+1$, where $a$, $b\in\mathbb{R}$.

$(1)$ If $a^{2}+b=0$,

$(1.1)$ When $a > 0$, find the extremum of the function $f(x)$ (expressed in terms of $a$);

$(1.2)$ If $f(x)$ has three distinct zeros, is there a real number $a$ such that these three zeros form an arithmetic sequence? If yes, find the value of $a$; if not, explain why;

$(2)$ The tangent line to the graph of $f(x)$ at point $A$ intersects the graph again at another point $B$. The tangent line at point $B$ is $l_{2}$, and the slopes of $l_{1}$ and $l_{2}$ are $k_{1}$ and $k_{2}$ respectively, with $k_{2}=4k_{1}$. Find the relationship between $a$ and $b$.

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