
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
If a function $f(x)$ defined on a domain $D$ satisfies the condition that for any $x \in D$, there exists a constant $M > 0$ such that $|f(x)| \leq M$ holds, then $f(x)$ is called a bounded function on $D$, where $M$ is called the upper bound of the function $f(x)$. Given the function $f(x) = \frac{1 - m \cdot g(x)}{1 + m \cdot g(x)}$, where $g(x) = (2 + x) \ln(1 + x) - 2x$.

1. Find the range of the function $g(x)$ on the interval $(\frac{1}{e} - 1, e - 1)$, and determine whether the function $g(x)$ is bounded on this interval. Provide a reason for your answer.
2. If the function $f(x)$ is a function with an upper bound of 3 on the interval $[1, e-1]$, find the range of values for the real number $m$.

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