
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
(The full score of this sub-question is 14 points)  
It is known that the function $f(x) = ax + \ln(x)$ with $a \in \mathbb{R}$.  
(1) If $a = 2$, find the slope of the tangent line to the curve $y = f(x)$ at $x = 1$.  
(2) Find the intervals on which $f(x)$ is monotonic.  
(3) Let $g(x) = x^2 - 2x + 2$, and for any $x_1 \in (0, +\infty)$, there exists $x_2 \in [0,1]$ such that $f(x_1) < g(x_2)$. Determine the range of values for $a$.

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