
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 $a = -1$, find the slope of the tangent line to the curve $y = f(x)$ at $x = \frac{1}{2}$;

(Ⅱ) Determine the intervals of monotonicity for $f(x)$;

(Ⅲ) Let $g(x) = 2^x - 2$. If there exists $x_1 \in (0, +\infty)$, such that for any $x_2 \in [0, 1]$, $f(x_1) \geq g(x_2)$ holds, find the range of $a$.

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