
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
The length of an interval $I=(α,β)$ is defined as $β-α$. Given the function $f(x)=ax^{2}+(a^{2}+1)x$, where $a < 0$, and the interval $I={x|f(x) > 0}$.

(I) Find the length of the interval $I$;

(II) Let the length function of the interval $I$ be $g(a)$, determine the monotonicity of the function $g(a)$ on $(-∞,-1]$;

(III) In the function $g(a)$ above, if $a∈(-∞,-1]$, does there exist a real number $k$ such that $g(k-sin⁡x-3)≤g(k^{2}-sin^{2}⁡x-4)$ holds for all $x∈R$? If so, find the range of $k$; if not, explain the reason.

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