
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 a function $f(x)$ defined on the domain $D$, if there exists a closed interval $[a, b] \subseteq D$ and a constant $c$ such that for any $x\_1 \in [a, b]$, $f(x\_1) = c$, and for any $x\_2 \in D$, when $x\_2 \notin [a, b]$, $f(x\_2) > c$ always holds, then the function $f(x)$ is called a "flat-bottomed" function on the domain $D$.

1. Determine whether $f\_1(x) = |x - 1| + |x - 2|$ and $f\_2(x) = x + |x - 2|$ are "flat-bottomed" functions on $R$, and explain the reasons.
2. If the function $g(x) = mx + \sqrt{x^2 + 2x + n}$ is a "flat-bottomed" function on the interval $[-2, +\infty)$, find the values of $m$ and $n$.

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