
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 $f(x)=ax^{2}+(3+a)x+3$, where $a\in \mathbb{R}, a\neq 0$,

$(1)$ If $f(2)=3$, find the expression of the function $f(x)$;

$(2)$ Under the condition of $(1)$, let $g(x)=f(x)-kx$, if the function $g(x)$ is monotonic in the interval $[-2,2]$, find the range of the real number $k$;

$(3)$ Is there an $a$ such that the maximum value of the function $f(x)$ in the interval $[-1,4]$ is $4$? If it exists, find the value of $a$; if not, please explain why.

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