
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 $f(x) = x^2 - a \ln x, (a \in \mathbb{R})$.
1. If the tangent line to the curve $f(x)$ at the point $(1, f(1))$ is perpendicular to the line $y = -x + 5$, find the value of the real number $a$.
2. If there exists $x_0 \in [1, e]$ such that $\frac{f(x_0) + 1 + a}{x_0} \leq 0$, find the range of values for the real number $a$.

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