
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
Let $f(x) = \frac{x^2}{2} + (1 - k)x - k \ln x$.

(1) If the tangent line to $f(x)$ at $(1, f(1))$ is parallel to $4x - y - 3 = 0$, find the value of $k$.

(2) Discuss the monotonicity of $f(x)$.

(3) If $k$ is a positive number and there exists $x_0$ such that $f(x_0) < \frac{3}{2} - k^2$, find the range of $k$.

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