
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 $f(x)=k\ln x-\frac{1}{x}$, where $k$ is a constant and $k\in \mathbb{R}$.<br/>$(Ⅰ)$ When $k=1$, find the equation of the tangent line to the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$;<br/>$(Ⅱ)$ Find the intervals where the function $f\left(x\right)$ is monotonic;<br/>$(Ⅲ)$ If the function is monotonically decreasing on $(2,+\infty)$, directly write down a valid value of $k$.

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