
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 $f(x) = x - \ln x - 2$.
1. Find the equation of the tangent line to the curve $y = f(x)$ at $x = 1$.
2. The function $f(x)$ has a root in the interval $(k, k+1)$ where $k \in \mathbb{N}$. Find the value of $k$.
3. Let $g(x) = \frac{1}{2}x^2 - bx - 2 - f(x)$. If $x_{1}$ and $x_{2}$ ($x_{1} < x_{2}$) are two extreme points of the function $g(x)$, and if $b \geq \frac{3}{2}$ and $g(x_{1}) - g(x_{2}) \geq k$ always holds, find the range of real numbers for $k$.

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