
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
Define the length of the interval I=(α,β) to be β-α. Given a function $f(x) = ax^2 + (a^2 + 1)x$, where $a<0$, and the interval $I=\{x \mid f(x) > 0\}$,
(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 the interval $(-∞, -1]$;
(Ⅲ) Given the function $g(a)$ mentioned above, for $a \in (-∞, -1]$, determine if there exists a real number $k$ such that $g(k-\sin x - 3) \leq g(k^2 - \sin^2 x - 4)$ holds for all $x \in \mathbb{R}$. If it exists, find the range of $k$; if not, explain why.

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