
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 $f(x) = ax + \ln x$ ($a \in \mathbb{R}$).  
(Ⅰ) If $a=2$, find the slope of the tangent line to the curve $y=f(x)$ at $x=1$;  
(Ⅱ) Determine the intervals of monotonicity for $f(x)$;  
(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 values for $a$.

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