
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 $f(x)$ defined on the domain $D$, if there exists a constant $M > 0$ such that $|f(x)| \leqslant M$ holds for any $x \in D$, then $f(x)$ is called a bounded function on $D$, where $M$ is the upper bound of the function $f(x)$. For the function $f(x)=1+x+ax^{2}$,
(1) When $a=-1$, find the range of the function $f(x)$ on $(-\infty,0)$ and determine whether the function $f(x)$ is bounded on $(-\infty,0)$, providing a reason;
(2) If the function $f(x)$ is bounded by 3 on $x \in [1,4]$, find the range of the real number $a$.

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