
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
(I) Given the function $j(x)= \frac{a}{x+1}$, where $a$ is a positive constant.

(1) If $f(x)=\ln(x)+j(x)$, and $a= \frac{9}{2}$, find the intervals of increase for the function $f(x)$;

(2) If the function $g(x)=|\ln(x)|+j(x)$, and for any $x_1$, $x_2 \in (0,2]$, $x_1 \neq x_2$, it always holds that $\frac{g(x_2)-g(x_1)}{x_2-x_1} < -1$, find the range of values for $a$.

(II) Given the function $f(x)=\ln(x)-ax+ \frac{1-a}{x}-1$.

(1) When $a \geqslant 0$, discuss the monotonicity of $f(x)$;

(2) Let $g(x)=x^{2}-2bx+4$, when $a= \frac{1}{4}$,

  ① If for any $x_1 \in (0,2)$, there exists $x_2 \in [1,2]$ such that $f(x_1) \geqslant g(x_2)$, find the range of values for the real number $b$;

  ② If for any $x_1$, $x_2 \in (1,2]$, it always holds that $|f(x_1)-f(x_2)|\leqslant l| \frac{1}{x_1}- \frac{1}{x_2}|$, find the range of values for $l$.

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