
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 a function $f(x)=kx^{2}+(3+k)x+3$, where $k$ is a constant and $k\neq 0$.
(1) If $f(2)=3$, find the expression of the function $f(x)$;
(2) Under the condition in (1), let $g(x)=f(x)-mx$. If $g(x)$ is a monotonic function on the interval $[-2,2]$, find the range of real number $m$;
(3) Does there exist a value of $k$ such that the maximum value of the function $f(x)$ on the interval $[-1,4]$ is $4$? If yes, find the value of $k$; if not, explain why.

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