
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 $f(x)=ax+\ln x$ where $a\in\mathbb{R}$.

(I) If $a=2$, find the slope of the tangent line to the curve $y=f(x)$ at $x=1$;  
(II) Determine the intervals where $f(x)$ is monotonic;  
(III) Let $g(x)=x^{2}-2x+2$. If for any $x_{1}\in(0,+\infty)$, there exists $x_{2}\in[0,1]$ such that $f(x_{1}) < g(x_{2})$, find the range of $a$.

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