
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
Consider the function $f(x) = (x - 1)^2 + a \ln x$, where $a \in \mathbb{R}$.
(1) If the tangent line of the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x + 2y - 1 = 0$, find the value of $a$.
(2) Find the monotonic intervals of the function $y = f(x)$.
(3) If the function $f(x)$ has two extreme points $x\_1$, $x\_2$ and $x\_1 < x\_2$, find the range of $f(x\_2)$.

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