
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 $f(x) = a(x-1)^2 + \ln x, a \in \mathbb{R}$.

(1) When $a = -\frac{1}{4}$, find the monotonic intervals of the function $y = f(x)$;

(2) When $a = \frac{1}{2}$, let $h(x) = f(x) - 3 \ln x + x - \frac{1}{2}$, find the maximum and minimum values of $h(x)$ on $[1, e]$;

(3) When $x \in [1, +\infty)$, all the points on the graph of the function $y = f(x)$ are in the region represented by the inequality system $\begin{cases} & x \geqslant 1, \\ & y \leqslant x - 1 \end{cases}$, find the range of the real number $a$.

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