
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 functions $f(x) = (2x + b)e^x$ and $F(x) = bx - \ln x$ with $b \in \mathbb{R}$.
1. If $b < 0$ and there exists an interval $M$ such that $f(x)$ and $F(x)$ share the same monotonicity on $M$, find the range of $b$.
2. If $b > 0$ and $g(x) = bx^2 - 2x - F(x)$ has a minimum value of $-2$ on the interval $[1, e]$, find the range of $b$.

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