
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 that $O$ is the coordinate origin, $P(x,y)$ is a point on the function $y=1+\ln x$, and the slope of the line $OP$ is $k=f(x)$.
(I) Find the range of values for the real number $m$ such that the function $f(x)$ has an extreme value in the interval $(m, m+\frac{1}{2}) (m > 0)$.
(II) Find the range of values for the real number $t$ such that the inequality $f(x) \geq \frac{t}{x+1}$ holds when $x \geq 1$.

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