
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 $f(x) = a(x - 1)^2 + \ln x + 1$,

(I) Find the extreme value of the function $f(x)$ when $a = -\frac{1}{4}$;

(II) Find the range of values for $a$ when all points on the graph of the function $y = f(x)$ are within the plane region represented by $\begin{cases} x \geq 1 \
y - x \leq 0 \end{cases}$ for $x \in [1, +\infty)$.

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