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

1. If $f(x) > 0$ holds for any $x \in (1, +\infty)$, find the range of values for $a$.
2. Under the condition of part 1, if the equation $f(x) + a + 1 = 0$ has exactly one real root in $x \in (0, 2]$, find the range of values for $a$.

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