
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 that $a$ and $b$ are real numbers, consider the functions $f(x)=x^{3}+ax$ and $g(x)=x^{2}+bx$. Let $f′(x)$ and $g′(x)$ be the derivatives of $f(x)$ and $g(x)$, respectively. We say that $f(x)$ and $g(x)$ have consistent monotonicity on the interval $I$ if $f′(x)g′(x) \geqslant 0$ always holds on $I$.
(1) Suppose $a > 0$. If the functions $f(x)$ and $g(x)$ have consistent monotonicity on the interval $[-1,+\infty)$, find the range of the real number $b$.
(2) Suppose $a < 0$ and $a \neq b$. If the functions $f(x)$ and $g(x)$ have consistent monotonicity on the open interval with endpoints $a$ and $b$, find the maximum value of $|a-b|$.

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