
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 $f(x)=(2x+b)e^{x}$, $F(x)=bx-\ln x$, where $b\in\mathbb{R}$,

(1) If $b < 0$ and there exists an interval $M$ such that $f(x)$ and $F(x)$ have the same monotonicity on interval $M$, find the range of values for $b$;

(2) If $b > 0$ and $g(x)=b{x}^{2}-2x-F(x)$ has a minimum value of $-2$ on the interval $[1,e]$, find the range of values for $b$.

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