
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 the function $f(x)=x\ln(x+1)+\left(\frac{1}{2}-a\right)x+2-a$, where $a\in\mathbb{R}$.

(I) When $x>0$, determine the intervals where the function $g(x)=f(x)+\ln(x+1)+\frac{1}{2}x$ is monotonic.

(II) When $a\in\mathbb{Z}$, if there exists $x\geq0$ such that the inequality $f(x)<0$ holds, find the minimum value of $a$.

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