
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 $f(x) = \frac{1}{2}ax^2 + 2x - \ln x$.  
(1) If $a = -\frac{3}{4}$, determine the monotonicity of function $f(x)$.  
(2) If the function $f(x)$ is monotonically decreasing in its domain, find the range of the real number $a$.  
(3) When $a = -\frac{1}{2}$, the equation $f(x) = \frac{1}{2}x - b$ has exactly two distinct real roots on the interval $[1, 4]$. Find the range of the real number $b$.

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