
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 function $f(x)$ is increasing on some interval $I$ within its domain $D$, and $y=\frac{f(x)}{x}$ is decreasing on $I$, then $y=f(x)$ is called a "weakly increasing function" on $I$.
(1) Please judge whether $f(x)=x+4$ and $g(x)=x^2+4x+2$ are "weakly increasing functions" on the interval $x \in (1, 2)$ and briefly explain your reasoning;
(2) If the function $h(x)=x^2+(m-\frac{1}{2})x+b$ (where $m$ and $b$ are constants) is a "weakly increasing function" on the interval $(0, 1]$, determine the conditions that must be satisfied by $m$ and $b$.

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