
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. If the function $f(x)$ has a root in the interval $(k, k+1)$ where $k$ is a natural number, find the value of $k$.
3. If the inequality $\frac{(x-m)(x-1)}{x} > f(x)$ holds for any positive real number $x$, find the set of positive integers for $m$.

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